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© Hodder Education Higher GCSE Maths for CCEA Answers 1

Chapter 1 The language of number

Exercise 1.1 (page 9)

1 a 16b 49c 81d 196

2 a ±3b ±8c ±10d ±15

3 a 1b 8c 1000

4 a 3b 4c 5

5 a 35

b 74 c 23 × 54

6 a 625b 24c 68

7 64

8 ±11

9 9

10 ±1

11 150

12 64

13 900

14 35

15 0.6

16 n = 12 is one example.

17 4 and 20

18 5

19 8, 9 and 10

20 2, 3, 5, 7, 89 and 641

21 13 and 14


1 a 52 × 17b 22 × 32 × 5c 33 × 7d 23 × 5e 2 × 32 × 11f 32 × 13g 2 × 3 × 5 × 7h 25 × 32

i 35

j 32 × 7 × 11k 25 × 53

l 22 × 33 × 5 × 13

2 a = 2, b = 4, c = 3


1 a 6b 12c 14d 18e 16f 22g 7h 1050

2 a 45b 40c 120

d 120e 144f 950g 630h 198

3 882

4 84

5 a 2 × 2 × 2 × 3 × 5 × 5 × 13 or 23 × 3 × 52 × 13

b 2 × 3 × 13 = 78

6 a 2 × 2 × 2 × 2 × 2 × 3 × 3 or 25 × 32

b 2 × 3 = 6

7 280 seconds or 4 minutes 40 seconds

8 24 students

2 Higher GCSE Maths for CCEA Answers © Hodder Education

Chapter 2 Whole numbers


1 a 1090b 9100c 1154d 123 982e 99 375f 4115

2 a 2999b 2073c 2228d 9024

e 853f 2876

3 a 3487b 4572c 4571d 3488

4 140 passengers

5 31 652 people


1 a 63b 24 000c 360d 400e 80 000f 540 000

2 a 12 000b 18c 80d 304 000e 600f 7440

3 a 3015b 13 122c 27 384d 140 213e 150 332f 267 344

4 a 54b 96c 54d 243e 451f 354

5 a 680b 230

6 a 22 236b 22 236 000

7 74 cartons

8 4380 miles

9 21 stamps, with 17p change

10 £12 370

Chapter 3 Decimals


1 a 88.947b 718.889c 85.774d 40.83e 59.22f 810.493

2 a 28.08b 0.136


c 22.68d 9.158e 0.0678f 205.8232g 5.423h 0.0057i 0.248j 20.832k 14.964l 3.5

3 a 12.6b 24.6c 18.3d 8.9e 0.2335f 200g 36.8h 56.8i 2.6j 12.5k 16l 65.4

4 a >b <c >d >e <f >

5 2.725 m

6 24.975 kg

7 10.03 seconds

8 £14.20

9 £2.04

10 0.055 cm

Chapter 4 Negative numbers


1 a −5b 1c 0d −19

e 6f 15g −8h −16i −4j −12k −10l 14m −4n −15o −18p 0

2 a 3b −4c 6d 2e 15f −9g 22h 0i −9

3 a 170 mb 15 m

4 £2050


1 a −15b −16c 35d 9e −5f 6g 18h 4i 16j 4

k −12

l 2m −36n 6o −48p −24

2 a −100b −2c 4


d 80e −144f 8g ±9h −4i −3

Chapter 5 Fractions


1 a45 =

1620

b811 =

2433

c 13 =

1751

d 58 =

2032

e 1016 =

58

f 2436 =

23

2 a 45

b 58

c 67

d 13

e 45

f 35

g 34

h25

i 67

j37

k78

l45

3 a 310

, 920

, 12, 4

5

b 25, 8

15, 11

20, 7

12

4 a 445

b 516

c 212

d 227

e 714

f 313

g 757

h 813

5 a 53

b 234

c 417

d 172

e 577

f 225

g 176

h 313


1 a 5712

b 31112

c 4415

d 11724

e 435

f 416

g 7512

h 71320

2 16

3 614 hours

4 110 litre

5 1710 miles


6 521

7 1920 tonnes


1 a 16

b 412

c 678

d 9e 3

f 514

2 a 49

b 23

c 258

d 113

e 10

f 127

Exercise 5.4 (page 47) 1 a 4.95 kg

b 5.6 mc £10.50

2 334 miles

3 310

4 11110 m2

5 16

6 £1.60

7 £9500

8 12.5 cm by 7.5 cm

9 £87.50

10 310

11 a x = 3, y = 8 is one

example.b Yes; any two numbers

whose product is 24.


Chapter 6 Approximation and estimation

Exercise 6.1 (page52)

1 a 5.68b 0.1c 56.35d 0.008e 4.51f 2.90g 4.0h 14.988i 0.04j 14.5k 23.66l 30.0

2 a 4b 4c 2d 3 (or 2 or 1)e 4f 5g 4 (or 3 or 2)h 4i 2j 4 (or 3 or 2 or 1)k 2l 3

3 a 2.35b 0.007c 7000d 0.009e 2.07f 20.0g 10.0h 0.007i 4.1j 700

k 0.05l 0.050


1 a 1621.32 (option B)b 315.4 (option A)c 456 (option B)d 4.1064 (option C)e 3.22 (option C)f 1317.5 (option C)g 27.6 (option B)h 35.2 (option A)

Note that for questions 2, 3 and 4 there are alternatives to some of the estimates given here.

2 a 35b 7c 320d 40e 600f 80g 0.5h 1000

3 a 200b 40c 5d 70e 30f 30g 4000h 81

4 a 600 kmb 16p per milec 20 minutesd 6 cme £14 400


Chapter 7 The order of operations


1 12

2 11

3 25

4 10

5 30

6 10

7 0

8 51

9 10

10 0

11 20

12 44

13 15

14 16

15 2.65

16 4.2

17 6

18 45.6

19 1

20 112

21 8

22 3

23 91112

24 11140

25 −512

Exercise 7.2 (page 59)Note that there are alternative correct solutions to some of these.

1 4 × 5 + 10

2 18 − 14 ÷ 7

3 20 − 4 × 4

4 5 × 3 + 12

5 12 + 8 × 9

6 41 − 10 × 2

7 21 + 40 ÷ 5

8 24 − (4 + 8 × 2)

9 12 + 4 − 1 × 6

10 20 + 6 × 8 − 10

11 20 ÷ (2 + 1 × 3)

12 20 − 30 ÷ 10 + 7


1 4.9

2 0.774

3 15.625

4 ±6.2

5 11

6 −9

7 8.55

8 205

9 6.73

10 3

11 1.25

12 51112


1 3000

2 17.47

3 24.89

4 3.800

5 11.67

6 4.132

7 10.45

8 1.606

9 14.96

10 15.26

11 1.020

12 2.720

13 0.4396

14 19.92

15 −2.688

16 1.627

17 3.164

18 9.251

19 144.0

20 10.87

21 5.235

22 1.794

23 7.804

24 7.492

25 2676

26 2.053


Chapter 8 Ratio


1 a 8 : 13b 4 : 5c 5 : 9d 2 : 3e 10 : 3f 2 : 5g 9 : 7h 2 : 3i 1 : 5j 1 : 5k 5 : 3l 4 : 15m 3 : 2n 130 : 9o 44 : 15

2 a 1 : 4

b 1 : 37

c 1 : 212

d 1 : 310

3 8 : 7

4 3 : 2

5 4 : 1


1 £250, £550

2 48°, 60°, 72°

3 5 : 3

4 £48, £60

5 1 m

6 144 cards

7 0.8 kg


1 a 2 kmb 16 cm

2 3.84 m by 4.8 m

3 81.6 cm

4 1 : 15 000

5 27 cm

6 1.875 kg

7 280 animals

8 200 g sugar, 40 g cocoa

9 a about 15 kgb about 14 pints

10 a 5 : 4b 5 : 4c 25 : 16


Chapter 9 Percentages and finance


1 a1150 = 0.22

b225 = 0.08

c 1920 = 1.45

d140 = 0.025

e 12000 = 0.0005

2 a3

100 = 3%

b45 = 80%

c 3500 = 0.6%

d 114 = 125%

e 58 = 621

2 %

3 a 26% = 0.26

b 4123 % = 0.416·

c 60% = 0.6

d 312 % = 0.035

e 22.2· % = 0.2·

4 1725

5 a25, 42%, 0.45

b 0.3, 31%, 13

c 711, 0.64, 70%

d 0.1%, 1

100, 0.1

6 77.7· %

7 3313 %

8 0.036

9 3500

10 115

11 a 85%b 6%c 12%d 85%


1 a £54b 144 gc 544d 52.5 ge 2 litresf £14.40g 102 minutes

(or 1 hour 42 minutes)h 1.5 m

2 a 42.4 mb 325.5 kgc 210 kgd 1416 m

3 £42.75

4 £13.25

5 a 0.765 kgb £1.02c 28 litresd 276 hectarese 4.5p (or 5p)f 44g 0.2 kmh 6.82 kgi £23.40j 19.2 miles

6 a £3.38b 41.36 mc £4387.50d 78.28 m

7 £11.28

8 £6080

9 3.22 kg

10 £118.94


1 80%

2 7.14%

3 75%

4 27.1%

5 2.2%

6 35.4%

7 a 15%b 32.25%

8 18%

9 513 %

10 0.53· %

11 3313 %

12 5.26%


1 £300

2 1480 students

3 £36

4 £340

5 £36 000

6 £235.52

7 280 ml

8 12 cm, 8 cm

9 50 questions

10 £84


1 £63

2 £4913


3 £29 866.88

4 7%

5 £2471.04

6 22.05%

7 15 years

8 £710

9 The bank is cheaper by £47.24.

10 13.5%

11 £51.78

Chapter 10 Reciprocals


1 a 223

b 114

c 35

d 27

e 813

f −15

gyx

h1k

i 1100

j −213

k −59

l3r2p

2 a 412

b427

c −25

dqp

e 221

f 559

g −145

h −59

3 a False – the reciprocal of 1 is 1.

b False – zero has no reciprocal.

c True – the reciprocal

of a is 1a and the

reciprocal of 1a is a.


d False – for example,

3 > 2 but 13 < 12.

Chapter 11 Standard form


1 The numbers in a, c, f and h are in standard form.

2 a 3.45 × 104

b 4.5 × 107

c 8.1 × 103

d 6.75 × 1010

e 5.4 × 10−6

f 7 × 10−5

g 6 × 104

h 3 × 10−3

i 1.24 × 10−1

j 8.675 42 × 107

k 3.4 × 10−9

l 9 × 10−1

3 a 15 670b 0.000 0456c 2000d 234 000 000e 156.78f 0.041g 25 000 000h 0.001 57i 0.0008j 31k 0.7l 6 200 000 000

4 a 3 × 108 m/sb 36 000 000 kgc 2.3 × 108

d 5 zerose 5.2 × 108 km2

f 2 × 10−10 mmg 9 460 000 000 000 000 mh 11 zerosi 2.65 × 108

j 0.000 000 001 m

5 a 1.12 × 102, 1.2 × 102, 2.3 × 104, 2.2 × 105

b 1.3 × 10−2, 2.13 × 10−2, 3.2 × 101, 2.3 × 103

c 1.25 × 10−4, 5.5 × 10−4, 2.45 × 10−3, 6.5 × 10−2


1 a 2.94 × 103

b 4.62 × 102

c 3.607 × 105

d 8.14 × 10−3

e 9.406 × 10−2

f 4.14 × 107

g 9.675 84 × 102

h 4.12 × 107

i 9.454 × 105

j 2.75 × 106

2 a 6.804 × 105

b 1.3 × 102

c 2 × 103

d 1.4 × 10−3

e 5.145 × 10−5

f 5 × 1013

g 6.0976 × 10−4

h 8 × 10−14

i 2.04 × 10−6

j 2.5 × 105

3 2.52 × 1012

4 2.592 × 1010 km

5 7.2 × 108 km

6 13 minutes

7 a 7.54 × 103 mmb 6.12 × 105 mm2

8 2.7648 m

9 8.92 × 107 people

10 7.3 × 1022 kg

11 71%

12 1.42 × 10−1 kg


1 See answers to questions 1 and 2 from Exercise 11.2.

2 a 9.072 × 1013

b 1.024 × 10−7

c 1.2 × 10−2

d 3.416 × 108

e 7.44 × 103

f 2.8 × 100

g 5.276 × 10−5

h 6.348 × 10−5

3 5 × 10−3

4 1.098 × 1021 m3

5 1.336 × 10−23 kg


6 a 1.245 13 × 1027 kgb 117 timesc 1.819 × 1027 kgd Venus

7 about 512 hours

8 5.83 × 1012 miles

Chapter 12 Rational and irrational numbers


1 a rational – a terminating decimal

b rational – the square root of a square number

c irrational – involves πd rational – a fraction in

the form ab

e rational – a recurring decimal

f rational = 35 – a

fraction in the form ab

g rational – the cube root of a cube number

h irrational – involves π

i rational – a recurring decimal

j rational – a terminating decimal

2 a irrational – the square root of a non-square number

b rational – a recurring

decimal = 4190

c rational – a terminating decimal

= 65 or

115

d rational – an integer

= 41

e rational – an integer

= 51

f rational – a terminating decimal

= 725

g rational – a mixed

number = 114

h rational – a terminating decimal

= −52 or −2

12

i rational – a terminating decimal

= 913200 or 4113

200j rational – an integer

= 21

3 a 3 is one example.

b 4 is one example.

43π2π or

12

3 are two

examples.

5 0.2 and 0.3· is one example.

0.2 = 15 is terminating and

0.3· = 13 is recurring but

0.2 + 0.3· = 15 +

13 =

815

which is rational.


6 rational

7 a rational – 614 = 25

4

= 52 – a fraction

b irrational – involves the square root of a non-square number

c rational – (1 12)2 = 9

4 –

a fractiond irrational – involves

the square root of a non-square number

e rational – π × 2 π = 2 –

an integer

f rational – ( 3)2 = 3 – an integerg rational – 0.23 = 0.008 – a terminating decimal

h rational – 2.50.2

= 12.5 –

a terminating decimal

i irrational – 414 = 17

4

– involves the square root of a non-square number

8 a irrational – involves πb rational – an integerc irrational – the square

root of a non-square number

d rational – a fraction

9 Yes; for example π and 5π.

Both these are irrational

but π × 5π = 5, which is

rational. Another pair of numbers is 2 and 50. Both these are irrational but 2 × 50 = 100 = 10 which is rational.

10 a 5 or 1

5

b π

c1

2 or 2

11 a 311

b523495 or 1

28495

c 411

d 169330


Chapter 13 Surds


1 15 2

2 7 3

3 18

4 5 2

5 2 13

6 12 2

7 12 10

8 40 2

9 100 10

10 54 2

11 9 5

12 6 11


1 a 6b 9c 2d 9

e 2 15

f 5

g 5

h 2

i 10 10

j 1

k 90 15

l 18 3

2 a 5

b 4 3

c 5 2

d 4 3

e 15

f 11 2

3 a 9 + 4 5

b 7 − 2 6

c 7 − 1

d 6

e 49 + 12 5

4 a = 54, b = −14

5 a irrationalb irrationalc rational

6 One possible value for x is 8.


1 a 2 3

b 2

c 13

d 4 2

e 3 2

f 2 3+3

–––––– 3

g 1 + 2 5

h 21+6

–––––– 3

2 a17 2

10

b25 2

2

c5 2

2 3 a false

b truec trued falsee truef false


Chapter 14 Algebra review


1 a −2ab 3x + 10yc −4k − 3d 0e 9abf 7p + 10qg 11t − 19rh 8e − 8e2

i −12a + 4kj 16p2 − 9pk 2a + 3b − 16l −8pq + 15m 11abc + 2bcn −k − 5t − 3tko 6a2b + 2a2 − 9b2

p −5t2 − 1q 8a2 + b2 − 9r −8w3 + 2w2

s 10 + 3pq + p2 − q2

t 10 − h3

u n6

v k7

w a2 × b4

x 28m3

2 expressions b and c

3 a 2a + 2bb ab

4 18a

5 a −3yb −2x2 + 7x − 2yc −6 − x − 4yd 5x


1 a 612

b −2

cx2 + 4

2 a 16b 20c 12d −8e 1f −1g 50h 100i 10

j −12

k −4l 5

3 a −2b −9c −32

d 14

e 10f 14g 2h −24i −4j −80

k −112

l −6

4 a −30b −36c −10d −6e 36f 44g −729

h 112

i 4j −3

12


k −6l ±6

5 634

6 −1 415

Chapter 15 Indices


1 a 24

b 32 × 43

c t 6

2 a x8

b p8

c a8

d t20

e n

3 a 8g5

b 54k9

c 3k4

d 6r 4

e4y4

3

f 25g4

g 8n6

h 4a6

i 108a5

j a7y5

k p2q5

l 27m6n9

4 a 26

b 52

5 a 1y2

b w7

c 1x2

d 1e y12

f 1

g a15

h x3

i x2

j x

k1

x5

l1

3 p8

m 1

n4

t

o k

6 a 1

b1

27m3

c 4w3

d 164c4

e 5y

f 18h3

g 3p

h 5y3

i 4k2

j 4xk 4nl p4q4

m t2

u4

n 3xy3

o 3b2c

7 a 1b 4a + b + c

c 2c + d − e

d1

4ey

8 a 37x

b 2b − 2c − a

9 a 36

b 214

c 81



1 64

2 3

3 1

4 116

5 81

6 32

7 1125

8 16

9 9

10 4

11 3

12 116

13 0.001

14 1

15 17

16 12

17 12

18 15

19 125

20 27

21 1

22 5

23 16

24 100

25 8

26 313

27 132

28 29

29 1008

30 16

31 343

32 a 12

b 312

c −13

33 2−6, 6−2, 612, 62, 26

34 160.25, 412 and 2

35 8

36 64

37 10052


1 5

2 −13

3 3

4 0

5 81

613

7 2

8 3

9 16

10 1

11 23

12 −112

13 −3

14 −1

15 32

16 −212

17 4

18 −1

19 1

20 10

21 −3

22 12

23 73

24 2

25 −112

26 2

27 6

28 113

29 −16

30 13


Chapter 16 Brackets


1 a 6x − 15b 20x − 8c 6a + 30d 8 − 6ae 15k − 5f 35d − 21g 24z − 16h 25p + 25i 12t + 20sj 6a − 4bk 1000 − 10xl 7a − 21c

2 a −8a + 12b −21c + 3c −25a − 10d −6t + 30e −3x − 3yf −12 − 8yg −4c − 7h −10a + 15bi −72y − 64j −3 + 9xk −5k − 5ml −10 + 100a

3 a x2 + 2xb 3x2 − xc xy − 2xd 2y2 + 3ye 6x2 − 2xf a2b + acg 3xy + 2y2

h −2a2 − ai a2b + ab2

j 8k2 − 12kk −3b2 − 18bl 6x3 − 21x2

4 a 6a + 17b 10t − 3c 14r − 15d 18a − 8e 18 − 12xf 6m − 9g 15a − 16h −b − 2i −t − 5j 22 − uk 10 − 3xl 15y − 7

5 a 5x2 − xb 6a2 − 3ac 8x2 − 10d 11y2 − 3y + 2e 8k2 − kf 1 − 4p2

g 2p − 4p2

h 6t 2 + 12ti 12a2 + 26aj −4xk xy2 − 2yl 17e 2 − 3e


6 a 26a + 16

b 23x − 9

c 10x − 12

d x + 8

e 11a − 20

f 31t − 4

g 7x − 15

h 5a2 + 7a

i 6x2 + 9x

j 6a − 12

k 3y − 2y2

l 5t 3 + 14t 2

7 x2 + 5x + 12 square units

8 (14y2 + 32y) cm2


1 a x2 + 6x + 8

b a2 + 8a + 15

c t 2 − 8t + 7

d y2 + 2y − 15

e u2 − 11u + 30

f p2 + 8p + 12

g a2 − 3a + 2

h c 2 − 9

i g2 − 16g + 63

j a2 + ax − ay − xy

k r 2 − 6r + 9

l c 2 − d 2

2 a 6x2 − x − 2

b 5a2 − 3a − 2

c 6t 2 + 19t + 15

d 12e 2 + 4e − 1

e 16y2 − 8y + 1

f 49k2 − 9

g 12b2 + 7b − 10

h 24 − 14d − 3d2

i 6a2 + ax − x2

j 18p2 − 27pq − 5q2

k 8 + 2x − 3x2

l 30y2 − 85y + 50

3 a x2 + 14x + 49b 16e2 − 24e + 9c 9d 2 + 12d + 4d x2 − 2x + 1e 4x2 − 4x + 5f 1 − 25x2 + 30xg 8y3 + 12y2 + 6y + 1h 36a2 − 78a + 36i 2x2 + 18j 3a2 + 4a − 7k x3 − x2 − x + 10l a3 − a2b − ab2 + b3

4 6x2 + 7x − 3

5 12a2 + 5a − 3

6 (4x2 + 140x) cm2

7 (x + 2)2 = (x + 2)(x + 2) = x2 + 2x +

2x + 4 (x − 2)2 = (x − 2)(x − 2)

= x2 − 2x − 2x + 4

So (x + 2)2 − (x − 2)2 = x2 + 2x + 2x + 4 −

(x2 − 2x − 2x + 4) = x2 + 2x + 2x + 4 − x2

+ 2x + 2x − 4 = x2 − x2 + 2x + 2x +

2x + 2x − 4 + 4 = 8x

8 n(n + 2)(n + 4) = n(n2 + 4n + 2n + 8) = n(n2 + 6n + 8) = n3 + 6n2 + 8n


Chapter 17 Linear equations


1 5

2 11

3 −3

4 2

5 23

6 −2

7 223

8 125

9 12

10 23

11 −1

12 −1

1337

1413

15 79

16 7

17 −16

18 −30

19 –56

20 212

21 −112

22 212

23 −1

24 34


1 −2

2 212

3 1

4 2

5 −3

6 45

7 −4

8 −2

9 0

10 −112

11 1

12 17

13 −12

14 56

15 8

16 2

17 2

18 0

19 1

20 −7

21 0

22 2

23 10

24 −1

25 37

26 −11


1 1

2 1

3 −512

4 115

5 −335

612

7 0

8 2

9 1

10 −4

11 3

12 3

13 1730

14 10

15 710

16 −137

17 29

18 3


1 6

2 10

3 60

4 4

5 1


6 −12

7 438

8 438

9 313

10 −514

11 −16

12 6

13 −18

14 1012

15 2

16 6

17 36

18 3

19 137

20 345

21613

22 −212


1 −118

2 142

3 −4512

4 3

5 −2

6 −2

7 4.5

8 −1


1 9

2 63, 65 and 67

3 4 packets

4 105°, 45° and 30° (x = 35°)

5 Jack 5, Martin 8, Rob 12

6 42 units

7 94.5 cm2

8 25 and 42

9 16 passengers

10 4 km (4000 m)

11 £24 000

12 10 cm, 8 cm and 6 cm

13 15 cm

Chapter 18 Sequences


1 a 6b 36


c 16d 20e 40

2 a add 5 to the previous term

b halve the previous term

c subtract 3 from the previous term

d multiply the previous term by 3

e subtract 1 more than was previously subtracted

3 a 42, 50b 41, 51c 20, 16d 55, 47e 47, 65f −4, −10g 11, 15

h 712, 3

34

i 29, 39j 56, 81

4 a 81, 130, 197b 37, 50, 65c 218, 345, 514d 222, 350, 520e 204, 329, 496f 37, 49, 63g 67, 93, 123h 211, 338, 507i 324, 539, 832j 134, 186, 246


1 a 4n − 3; 77b −2n + 8; −32c 12n; 240d 7n − 2; 138e 1.5n + 0.5; 30.5f −3n + 16; −44g 5n + 2; 102h −n + 11; −9i −5n + 105; 5j −2n − 1; −41

2 a n2 + 1; 101b n2 − n + 1; 91c 3n2 − 2; 298d n2 + 2n + 3; 123e −n2 + 4n + 1; −59f n2 − 2; 98g 5n2 + n − 3; 507h 20n2; 2000i n2 − 1; 99

j −12n2 +

12n + 10; −35

3 2n + 4; 34

4 3n − 1; 299

5 2n2 + n − 1; 299

6 17th term

7 24th term

8 The second difference is 2 so a = 1.

The original sequence is 2 6 12 20 30Subtracting the sequence n2

results in the linear sequence

−1 4 9 16 36––––––––––––– 1 2 3 4 5

This sequence follows the rule n2 + n, or n(n + 1).

Alternatively, having found a = 1, working back to n = 0 gives c = 0.

1st term = a + b + c = 1 + b + 0

But the 1st term is 2, so 1 + b = 2, and so b = 1.

Hence the rule for the nth term is n2 + n, or n(n + 1), as before.


1 a 18 pegsb 24 towels

2 (1000 − 0.5t) cm3

3 a 8 chairsb C = 2T + 2

c 26 chairsd 47 tables

4 a P = 4n − 3b T = 3n + 5

5 an

n + 1; 2021

b n × n2 or n3; 20 × 202 = 203 = 8000

c2n

n2 + 1; 40401

6 9 bricks

7 465

8 a12n2 − 1

12n

b 54 diagonals

9 a True; the number of chords forms a quadratic sequence with the given rule for the nth term.

b False; for 6 or 7 points, the number of regions does not follow the given rule for the nth term. For example, 6 points creates 31 regions but 26 − 1 = 25 = 32.

10 No; the sequence 10, 12, 14, 16, 18, 20, ... is a suitable counter example.


Chapter 19 Factorisation


1 a 4(x + 2)b 4(4 + n)c 3(4k + 3)d 5(2x − 3)e 3(7 + 2p)f 8(5x + 3)

2 a 12(2a − 3)b 4(x − 5)c 5(n + 1)d 4(3 + 4p)e 8(1 − k)f 7(x + 3)g 8(1 + 2y)h 3(3x − 1)i 8(4 − 5n)j 6(3y + 4)k 5(5t − 8)l 6(g − 3)

3 a 2x(2x + 1)b a(y − b)c h(2h + 1)d 3x(2x + 1)e 4a(a − 2)f 4pq(2p + 1)

4 a 5n(3 − 4n)b k2(k + 1)c 5a(2a − 1)d ab(a + b)e 2r2(4 − 5r)f 6a(3 − a)g πr(r + 2h)h 6cd(2c + 3d)i 2xz(y + 2x)j 5p2(6p − 5)k xy(x + w)l 2k2(2k − 5)

5 a m(m + 4)b 2e(8 − e)c x(6 − x)d 7x(8 + 3x)e r(4r − 1)f 5x(2 − 5x)g p2(p − 3)h 3y(5 − 3y)i πr(r + 2)j ab(b − c)k 4a2(2a + 1)l 2q(2p + 1)

m 2c(2a2 + 3)n ad(a2 − d)o a2b(5a + 2)p 2x(3y + z − 2w)q a(wx + ax + a2y)r 2xy(3y + 2 − y)


1 (b + a)(y + z)

2 (x − w)(y − z)

3 (x2 + y)(a + b)

4 (e + d )(f + g)

5 (x + y)(x + a)

6 (c − 1)(d 2 + 1)

7 (5d − 2f )(3c + 2e)

8 (2y − 1)(x + 3)

9 (1 + t 2)(1 + t)

10 (y − x)(a + b)

11 (a + 2b)(3y + x)

12 (u − 5)(4u − v)

13 (e2 + 2)(e − 3)

14 (3g + n)(2g − h)

15 (c − 5e)(3d + 2f )

16 (4p2 − 3t)(6 − t)

17 (a − 5)(k + 1)

18 (x − 1)(y2 + 1)


1 a (c − d )(c + d )b (a − y)(a + y)c (10 − p)(10 + p)

d (c − 1d)(c +

1d)

e (6y − 1)(6y + 1)f (8 − t)(8 + t)g (3k − m)(3k + m)h (4a − 7b)(4a + 7b)i (9a − 5)(9a + 5)

2 a 135b 2.4


c 8d 128 016

3 a (y − z)(2x + y + z)b (3n − 1)(5n − 1)c (x + 1)(5x + 3)

4 a 9999 = 1002 − 12

b 9999 = 99 × 101 = 32 × 11 × 101


1 a (x + 9)(x + 5)b (x + 5)(x − 7)c (x − 5)(x + 4)d (a − 10)(a − 3)e (p + 2)(p − 15)f (y + 8)(y + 3)g (a + 3)(a − 7)h (k + 1)2

i (x + 3)(x − 8)j (p − 1)(p − 9)k (m − 5)(m + 7)l (t + 4)(t − 9)

2 a (3a + 2)(a + 5)b (3x − 8)(x + 1)c (2x − 3)(x + 1)d (3y + 11)(y + 1)e (2p − 1)(p − 5)f (3x + 4)(x − 5)g (3p − 1)(p − 10)h (5y − 4)(2y + 1)i (3x + 4)(2x − 5)j (4y − 1)(3y + 4)k (4n − 1)(n + 3)l (5x − 2)2

3 a x(5x − 3)b (a − 11)(a + 11)c (2x − 3)(x + 1)d (x + 3)(x − 7)e (p − 3)(p + 3)f (3x − 2)(x − 3)g 5x(2x − 1)h (5x + 2)(2x + 1)i (7x − 6)(7x + 6)j (4p + 1)(2p − 3)k 4p(1 − 2p)l (y − 2)(y − 3)


1 a 5(a3 + 3)

b (1 − 8y)(1 + 8y)c (q + s)(p + r)d (x − 1)(x + 5)e (5y − 2)(3y + 1)f (3p + 1)(p − 6)g x(x2 + y)h (x − k)(x + y)

2 a 2(x + 1)(x − 3)b 2(x − 3)(x + 3)c 2(2y − 1)(x + 3)d 5(x + 4)(x − 3)e 4(a − b)(a + b)f 10ab(a − 4b)g 2(k − 2)(k − 3)h 4(p − 2)(q + 2)

3 a2b3a

b k – 2k + 3

c a + b4b

d 3(2a + 1)(3a + 1)

4 2y + 5x

5 208

6 Area = π(202 − 182) = π(20 − 18)(20 + 18) = π × 2 × 38 = 76π cm2


Chapter 20 Formulae


1 312 m

2 104 °F

3 −313

4 1.8 × 1014

5 338 350

6 2435

7 80 m

8 37.5

9 8.98

10 283 cm2

11 112

12 a W = 160.40 + 5.5gb £242.90

13 P = 100(S – C)

C


1 aPk

b x = c + t

c x = e + d

c

d x = c – d – b

ae x = g − f

f x = y + 3

4

g x = t + u

3

h x = w – 2y – 3z

yi x = Ny − y

j x = D – E

t

k x = p2 – q2

3l x = Q – T

P 2 a y = z − k

b y = ba

c y = d – a

t

d y = 4 − k

e y = L – H

G

f y = a – d

c

g y = A – B

M

h y = 3 – e – f

t

i y = h + j

j y = zx

k y = 5 – r

t

l y = Abc

3 a r = Aπl

b r = C2π

c r = d2

4 a = v 2 – u2

2s

5 T = pVR


1 a a = xy – b

x

b a = D – BC

C

c a = y – hn

h

d a = b + 5

5

e a = c – kt

k

f a = 6 – r

6

g a = g + de

d


h a = qn – px

p

i a = L – 2l

lt

j a = 100p – T

100

k a = P – 2b

2

l a = 5m – b3

2 a b = 4Aπ

b b = ma – Ft

m

c b = ca

d b = 2E – VAV

e b = 100(k – e)c

f b = fT

g b = 4I3m

h b = 5a – 40t

ti b = 2ah − a

j b = 3L + c

8

k b = T – a

T

l b = Q – 5PA

5P

3 h = 3Vπr 2

4 h = A – 2πr 2

2πr

5 n = T – a + d

d


1 a a = ± c 2 − b2

b a = ± c − 5

c a = ±2 b

d a = d 2

e a = b2 − x

f a = tb

g a = k2 + p

h a = ± bde

i a = 2st2

j a = 3hg

k a = b( c2π )2

l a = m2

n2

2 a x = ± C

b x = 3

D

c x = ±tk2

d x = (E − T)2

e x = (dc )2

− a

f x = ± q − p

g x = ( r2π)2

− t

h x = c − a2

i x = 3

w − y

j x = 33rπ

k x = 9

4k2

l x = p

3t

3 b = ± ac

4 r = A4π

5 l = ( T2π

)2 g


1 a x = c

a + b

b x = ef

d – e

c x = f

2g – h + k

d x = y – 1y + 1

e x = c – 3a3b + 1

2 a y = u

t + 4

b y = qn

2q + p

c y = m + na – 1

d y = –2

6 – p

e y = 100b

x + 100

3 n = IR

E – Ir

4 y = an

p – a


Chapter 21 Proportion and variation


1 £2.88

2 3150 km

3 2212 m

4 12 bottles

5 2 hours 24 minutes

6 15 days

7 28 hours

8 a 30 CDsb 16 200 CDs

9xn

n + 5

10 £14p/11


1 P = 3.2Q

2 a y = 3 xb 18

3 4000 cm3

4 a 8 amperesb 6.25 volts

5 15 m/s

6 a 9b 7

7 a a = 1.25 b − 3b 6.24c 2.5

8 1411.2 m

9 y ∝ x3

10 a p is multiplied by 4b p is divided by 16

11 a H = 36I2tb 4608 Jc 4 amperes


1 T = 32M

2 a y = 36x

b 16

3 720 cm3

4 a 7.5 amperesb 1.8 × 105 ohms


5 32 units

6 a 0.8b 11.6

7 350 m

8 a A = 3

B − 4b 3

4

c 4.04

9 25 mm

10 a 75 swings per minuteb 0.36 m

11 0.22 mm

12 a The force is divided by 8.

b The force is 200 times greater.

Chapter 22 Trial and improvement


1 a 5.7b 1.5c 4.7d 2.3e 3.5f 3.8g 3.7h 5.9i 4.1j 3.1

2 a 3.11b 6.58c 4.62d 1.62e 2.57f 3.31g 1.06h 2.45i 3.19j 2.64


1 6.4

2 5.88 cm

3 13.5 cm

4 a w 2 + 6w − 16b w = 6.2 (1 d.p.)

5 a 10 mb 22 mc 21 md 4.6 seconds

Chapter 23 Algebraic fractions


1 a25

b78

c78

2 a 2

ba2

c2q3

d7a12c

e25t

f a

3 a 3a + 1a


b 2p + 1p

cx + 1

x

dx + 2y

3xy

e 2(p + 2q)p

f 4 + 5k5

gm

m – n

he – 2e + 3

i3y – 1

3

jab

k2(t – 3)(t – 1)

la

a + 7

4 aa

a + 2

bp

p + 1

ca

a – 3

dx + 1x + 2

e h + 2h – 5

fk – 2

k

gy + 32y + 1

ha – 4

2

ib

3 + 2a2

ja + 4

2(a – 5)

k −ab

la – y

y


1 a5x8

b3a4

c5a8

d23p20

e13x12

f5x + 4

6

2 a7

10x

b16x

c132x

d8x – 9

3x2

e2p + 3

p2

f13c3d

3 a5x + 18

12

b 5p – 19q6

c–t – 9u

20

d3a + 8

10

e7x + 15

10

f11x + 8

12

4 a7a + 5a(a + 1)

b2a – 7

(2a – 1)(a – 2)

c13 – 2x

(x + 1)(x – 2)

dt(t – 3)

(t + 1)(t – 1)

e31k – 5

30k2

f 1

5 a2a2 + 7a – 7(a – 2)(a + 1)

by – 5

(y + 1)(y – 1)

c4a2 + 9

(2a + 3)(2a – 3)

d7p – 1

p(p + 1)(p – 1)

610 – 200x(x – 4)


1 a 4b

b5c3d

c 2p2q

dab

ex2y15

f4(a – b)

5

2 a4(a – b)

3

b1

2(x – 3)c 1d 2(p + 1)

e k – 2k(k + 2)

f 112

3 a 334

b 3 − 2x

cp

8(p – 1)

d(x + 1)(x + 2)

2e 2

fx(x + b)

2

4 a 135

bp + q

15

ca – 2

a

d8z

3y(x + 2y)


Chapter 24 Quadratic equations


1 a x = −3, x = 4b a = 7, a = −1

c x = −112, x = 2

d t = −114, t = −3

13

e x = 3, x = −3

f x = 12, x = 3

g x = 0, x = 114

h a = 0, a = 12

i x = 0, x = −2

j x = −12

2 a x = −2, x = −1

b y = −12, y = 3

c x = −13, x =

12

d x = 13, x = 1e a = −2, a = 4f p = 8, p = 3

g y = 13, y = 3

h k = −114, k = 2

i a = 34, a = −

12

j m = 34, m = −11

3

3 a x = 13, x = −

13

b x = 0, x = 12

c a = 0, a = −13

d x = 5, x = −5

e y = 0, y = −112

f y = 0, y = 112

g a = 12, a = −

12

h y = 0, y = −1i t = 3, t = −3

j x = 0, x = 412

4 a x = 2, x = 3

b x = 112, x = −5

c a = 212, a = −2

d x = ±1e x = 1, x = 3f a = −3, a = 11g p = 4, p = 5h x = −1, x = 6i k = 6, k = −10j n = 0, n =

16

5 a x = −312, x = 2

b x = 12, x = −3

c a = 2, a = 6

d a = 34, a = −1

12

e x = 0, x = −112

f n = ±123

g x = 0, x = 1

h x = −56, x = 1

i a = 2, a = −5j a = 0, a = 1

25


1 a x = 3.73, x = 0.27b x = 0.44, x = −3.44c x = 0.85, x = −1.18


d x = −0.61, x = −4.89e y = 1.43, y = 0.23f x = 1.13, x = −0.88g x = 0.53, x = −1.65h x = −0.28, x = 3.61i x = 0.64, x = −1.00j y = 2.14, y = −0.47

2 a x = 1 ± 6

b x = 3 ± 7

2

c y = 6 ± 31

d x = −3 ± 3

2

e k = −5 ± 33

4

f x = −3 ± 2 2

g x = 6 ± 15

7

h y = 2 ± 11

i t = 3 ± 3

j a = −5 ± 17

8

3 a x = 2.85, x = −0.351b x = 1.45, x = −3.45c x = 4.27, x = −3.27d y = 0.452, y = 1.55e x = 1.61, x = −3.11f x = 2.47, x = −0.135g y = −2.59, y = −5.41h a = 1.64, a = −0.24i x = 7.87, x = 0.127j a = 3.41, a = 0.586


1 x = 521

2 a = 2

3 x = 3

4 a = −1, a = 8

5 p = 4, p = 2

6 a = −2, a = 6

7 x = 2, x = −4

8 p = −2, p = 5

9 x = 0, x = 3

10 x = −412, x = 5

11 y = 2.49, y = −1.69

12 a = 0.63, a = −2.38


1 32 and 35

2 24p

3 2.5 cm by 8.5 cm

445

5 29 and 31

6 2 discs

7 45 km/h

8 3.14 seconds

9 18th term

10 22 people

11 24th triangular number

12 30.73 m

13 35 cm

14 24 cm and 56 cm

15 18.5 cm

16 5 km/h

)


Chapter 25 Straight lines and linear graphs


1 a

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

–5

–6

–7

–8

2

3

4

5

6

7

8y = 2x – 1

y

x −3 −2 −1 0 1 2 3 4

y −7 −5 −3 −1 1 3 5 7

b

2 3 4 5 x1–1–2–3–4

2

–2O

–4

–6

–8

4

6

8

10

12

14

16y = 3x + 2

y

x −3 −2 −1 0 1 2 3 4

y −7 −4 −1 2 5 8 11 14c

18

2 3 4 5 x1–1–2–3–4

2

–2O

–4

–6

–8

–10

–12

–14

4

6

8

10

12

14

16y = 4x

y

x −3 −2 −1 0 1 2 3 4

y −12 −8 −4 0 4 8 12 16

d

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4

5

y = –x + 1

y

x −3 −2 −1 0 1 2 3 4

y 4 3 2 1 0 −1 −2 −3e

2 3 4 5 x1–1–2–3–4

2

–2O

–4

–6

–8

–10

–12

–14

–16

–18

4

6

8

10

12

14

16

18

20

y = 5x – 2

y

x −3 −2 −1 0 1 2 3 4

y −17 −12 −7 −2 3 8 13 18


f

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

–5

–6

–7

–8

2

3

4

5

6

7

8

y = –2x + 1

y

x −3 −2 −1 0 1 2 3 4

y 7 5 3 1 −1 −3 −5 −7

g

2 3 4 5 x1–1–2–3–4

1

O

2

3

4

5

6

7y

12y = x + 4

x −3 −2 −1 0 1 2 3 4

y 2.5 3 3.5 4 4.5 5 5.5 6

h

2 3 4 5 x1–1–2–3–4

1

–1O

2

3

4

5

6

8

7

y

y = 4 – x

x −3 −2 −1 0 1 2 3 4

y 7 6 5 4 3 2 1 0

i

2 3 4 5 x1–1–2–3–4

2

O

4

6

8

10

12

16

18

14

y

y = 10 – 2x

x −3 −2 −1 0 1 2 3 4

y 16 14 12 10 8 6 4 2

j

2 3 4 5 x1–1–2–3–4

1

O

2

3

4

5

6

8

9

10

7

y

y = 8 – x12

x −3 −2 −1 0 1 2 3 4

y 9.5 9 8.5 8 7.5 7 6.5 6

k

2 3 4 5 x1–1–2–3–4

2

–2O

–4

–6

–8

–10

–12

–14

4

6

8

10

y = – 3x

y

x −3 −2 −1 0 1 2 3 4

y 9 6 3 0 −3 −6 −9 −12

l

2 3 4 5 x1–1–2–3–4

1

O

2

3

4

5

6y

y = x + 62

x −3 −2 −1 0 1 2 3 4

y 1.5 2 2.5 3 3.5 4 4.5 5

2

2 3 4 q1–1–2–3–4

1

–1O

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

2

3

p = 2q – 4

p

3 a

2 3 4 x1–1–2–3–4

2

–2O

–4

–6

–8

–10

–12

–14

–16

4

6y = 3x – 6

y

b (2, 0)c (0, −6)


4 a Nob Noc Yes

5 y = x + 3, y = 2x + 5,

y = 12 x + 2 and x + 2y = 0

6 (−2 12, 0)

7 2

8 a 7

b −12

c −13


1 a 6 5

b 5 2

c 4 10

d 74

e 4 2

2 a (4, 312)

b (2, 4)c (−3, −4)d (−1, −7)

e (12, −1) 3 B(−3, 6)

4 Yes

5 4, 13, 13; isosceles since two sides are equal

6 22 + 52 = ( 29)2, satisfies Pythagoras’ theorem − right-angled


1 L1= –73 L2 = − 4

11

L3 = 0 L4 = −15

L5 = 1 L6 = ∞

L7 = 2

2 PQ = 2 QR = 12

RS = 7 SP = −34

3 a 13

b 2

c 53

d −52

4 a q = −8b p = 2

5 a = 5 and b = 2 Yes, there is more than

one possible set of values, e.g. a = 8 and b = 7.

a and b can take any values

such that b + 3a – 2 simplifies

to 53.

6 37.5 m

7 52

8 y = 0


1 a gradient = 2, y-intercept (0, −7)

b gradient = 13,

y-intercept (0, −1)c gradient = 6,

y-intercept (0, 0)d gradient = 2,

y-intercept (0, −3)

e gradient = −13,

y-intercept (0, 3)f gradient = 2,

y-intercept (0, −12)

g gradient = −12,

y-intercept (0, 2)

h gradient = −23,

y-intercept (0, 13)

2 a

2 3 4 x1–1–2–3–4–5

1

–1O

–2

–3

–4

2

3

4

5

6

y = x + 4

y

b

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5

y = x – 3

y

c

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5

y = 3x + 1

y


d

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

–6

1

2

3

4

y = 2x – 5

y

e

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5y

12y = x + 1

f

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5y

y = –x + 4

g

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5y

2y = 5x – 2

h

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5y

13y = – x

i

2 3 4 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3

4

5

2y – x = 4

y

j

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5y

4x – 2y + 1 = 0

k

2 3 4 5 x1–1–2–3–4–1O

–2

–3

–4

–5

1

2

3

4

5y

3x – 2y = –6

l

2 3 4 5 6 x1–1–2–3–1O

–2

–3

–4

–5

1

2

3

4

5y

2x + 5y = 10


3 k = ± 2

4 y = 0

5 a 4x + 12y = 6 and x + 3y = 6 are parallel or y = 4 − 5x and 5x + y = 10 are parallel.

b 2y = x − 4 and y = 5 − 2x are perpendicular or y = −x + 6 and y = x − 1 are perpendicular.

6 a = −12


1 line L1, y = 8line L2, y = x + 6

line L3, y = −12x + 1 or

2y + x = 2

line L4, y = 32x − 4 or 2y =

3x − 8

line L5, x = −8

line L6, y = −12x + 5 or

2y + x = 10

2 line A, y = −2x line B, x = 3

line C, y = −13x + 7 or

3y + x = 21

line D, y = 32x + 3 or

2y = 3x + 6

line E, y = −5

line F, y = −53x − 10 or

3y + 5x + 30

= 0

3 y = −2x + 3

4 y = −3x − 2

5 y = −12x or 2y + x = 0

6 y = 3x − 5

7 y = −12x − 1 or 2y + x + 2

= 0

8 y = 3x − 2

9 y = −13x + 4 or 3y + x

= 12

10 y = −14x − 2 or 4y + x + 8

= 0

11 y = −32x + 4 or 2y + 3x

= 8

12 y = −34x + 9

12 or 4y + 3x

= 38


Chapter 26 Non-linear graphs


1

1 2 3 x–1–2–3–4–5 O

–2

–4

2

4

6

8

10

12y = x2 + 3x

y

2

1 2 3 4 x–1–2–3–4 O

–2

2

4

6

8

10

12

14

16

y = x2 – 2x

y


3

1 2 3 4 5 x–1–2–3 O

–8

–6

–4

–2

2

4

6

8

y = x2 – 3x – 4

y

4

1 2 3 x–1–2–3–4–5 O

–2

–4

–6

–8

2

4

6

8

10

12

14

16

y = 2x2 + 3x – 6

y


5

1 2 3 4 5 6 7 xO

–2

2

4

6

8

10

12

14

16

18

y = x2 – 8x + 15

y

6

1 2 3 4 x–1–2–3–4–5–6 O

–8

–10

–6

–4

–2

2

4

y = 1 – 3x – x2

y


7

1 2 3 4 x–1–2–3–4 O

–8

–10

–12

–14

–16

–6

–4

–2

2

4

6

8

y = 6 + x – 2x2

y

8

1 2 3 4 x–1–2 O

2

4

6

8

10

12

14

16

18y = 3x2 – 5x + 4

y


9

1 2 3 4 5 x–1–2 6O

–8

–6

–4

–2

2

4

6

8

y = x(5 – x)

y

10

1 2 3 4 x–1–2–3 O

–6

–4

–2

2

4

6

8

10

12

14

y = (x + 1) (2x – 3)

y



1

1 2 3 x–1–2–3 O

–20

–30

–10

10

20

30

40

y = 6 – x3

y

2

1 2 3 x–1–2–3 O

–40

–60

–20

20

40

60

y = 2x3 + 3

y

3

1 2 3 x–1–2–3 O

–20

–30

–40

–10

10

20

30

y = x3 – 4

y

4

1 2 3 4 x–1–2–3–4 O

–20

–30

–10

10

20

30

y = x3 – 9x

y


5

1 2 3 x–1–2–3 O

–20

–30

–40

–10

10

20

30

y = x3 + x – 3

y


1

2 4 6 8 x–2–4–6–8 O

–8

–6

–4

–2

2

4

6

8y

y = 6x


2

2 4 6 8 10 12x–2–4–6–8–10–12 O

–8

–10

–12

–6

–4

–2

2

4

6

8

10

12y

y = 20x

3

1 2 3 4 x–1–2–3–4 O

–4

–3

–2

–1

1

2

3

4y

y = –3x

4 x = −2



1 a

1 2 3 4 x–1–2–3–4 O

2

4

6

8

10

y = 2x

y

b x = 2.3

2 a

0.5 1 1.5 2 x–0.5–1–2.5–2 O

2

4

6

8

10

y = 1 + 3–x

y

b y = 1


1

60 120 180x(°)–60–120–180 O

–1

–0.5

0.5

1

y = sin x

y

2

240180 300 360x(°)12060O

–1

–0.5

0.5

1

y = cos x

y

3

270180 360x(°)90O

y = tan x

y

4

60 120 180x(°)–60–120–180 O

–2

–1

1

2

y = 2 sin x

y

5

240180 300 360x(°)12060O

–2

–3

–1

1

2

3y = 3 cos x

y


6

–180–270 –90–360 x(°)

y = tan x

y

For the sine and cosine curves for questions 7 and 8, refer to pages 260 and 261 of the textbook.

7 a i 0.64ii 0.98iii−0.34iv −0.77

b i 53°, 127°ii 217°, 323°

8 a i 0.64ii −0.64iii−0.34iv0.64

b i 73°, 287°ii 134°, 226°


Chapter 27 Real-life graphs


1 a u = 5, a = 10b 22 seconds


2 a

50 60T30 402010O

20

21

22

23

24

25

26L

b Yes; L = 0.1T + 20c 27.5 mm

3 a T = −3.75m + 30b 6 minutes 40 seconds


1 a

5 63

Mass m (kg)

t = 3m + 12

Coo

king

tim

e t (

hour

s)

421O

1

2

3

4

5

6

7

b 412 hours

c 1.5 kg


2 a

10 12 14 166

Distance (miles)

Fare

(£

)

842O

5

10

15

20

25

30

b 6 milesc £2.50

3 a Claireb Claire 16.36 mph, Dermot 10 mphc 13.30d They are both 10 miles from their destination.


1 a

x 0 2 4 6 8 10

10x 0 20 40 60 80 100

−x2 0 −4 −16 −36 −64 −100

y 0 16 24 24 16 0

b

10 12 x6

y = 10x – x2

842O

5

10

15

20

25

30y

c 25 m


2 a

5 63

t (seconds)

h (m

etre

s)h = 30t – 5t2

421O

10

20

30

40

50

b 3 secondsc 45 m

d 312 seconds

3 a

t 0 1 2 3 4 5 6t2 0 1 4 9 16 25 36−9t 0 −9 −18 −27 −36 −45 −5410 10 10 10 10 10 10 10P 10 2 −4 −8 −10 −10 −8

b

2 4 6 8t hours

O

–2

–4

–6

–8

–10

–12

2

4

6

8

10

12

P = t

2 – 9t + 10

P (°

C)


c 10 °Cd −6.25 °Ce 1 hour 18 minutes


1 a

2040 2050 20602020 2030

Year20102000O

30

40

50

60

Popu

latio

n (m

illio

ns)

70

80

90

b during the year 2023

2 a £10 200b £8670c £12 000 × 0.85t

3 25 years


28 Simultaneous equations


1 a x = 4, y = 4b x = 9, y = 5c x = 1, y = −2

2 a x = 2, y = 3b x = 2, y = 3c x = 2, y = −1

d x = 12, y = 3

3 a x = 2, y = 3b x = 6, y = 1c x = 5, y = 5d x = 2, y = −3

4 (0, 3), (2, 5), (5, 5) and (3, 1)


1 x = 9, y = 1

2 a = 2, b = −1

3 p = 112, q = −2

12

4 e = 3, f = −12

5 x = 4, y = 6

6 x = −1, y = 3

7 x = 1, y = 2

8 m = 4, n = −1

9 x = 212, y = −5

10 x = 1, y = 3

11 x = 5, y = −3

12 x = −5, y = 4


1 a x = 112, y = −2

12

b a = 9, b = 4c c = −1, d = 3d p = 2, q = −3e x = 7, y = 1f a = 5, b = −4

2 a x = 0, y = −1b a = 9, b = 1c p = −2, q = 5d a = 13, b = −7e x = 3, y = 1

f p = −234, q = 11

4

3 a x = 3, y = 2b a = 3, b = 4c x = 3, y = 3d p = 7, q = 2

e a = −235, b = 3

25

f p = 3, q = −3

4 a x = 2, y = 1

b x = 3 310

, y = 115

c a = 3, b = 4d x = 2, y = −2e x = 2, y = −4f x = 3, y = 2



1 7 and 3

2 rubber 6p, protractor 25p

3 magazine £1.40, newspaper 72p

4 length 12 cm, width 8 cm

5 adult £17.50, child £12

6 man 37 years, son 7 years

7 11 years old

8 first number 37, second number 24

9 1500 books sold during promotion, 3500 books after

10 £6.40

11 42 and 24

12 y = −3x + 5

13 39 cm


1 x = 1, y = 1

2 x = −3, y = 2 or x = 4,

y = 413

3 x = 5, y = 1 or x = −7, y = −5

4 x = 6, y = 2 or x = −4, y = −3

5 x = −3, y = 0 or x = 4, y = 7

6 x = 1 ± 3, y = 5 ± 3 3

7 x = −1 ± 6, y = −3 ± 2 6

8 x = −3 ± 69

5,

y = 1 ± 3 69

5

9 x = −5 ± 57,

y = 3 ± 57

4

10 x = 9 ± 4 14

11,

y = 3 ± 6 14

1111 (2, 0) and (3, −1)

12 (312, 5) and (−3, 18)


29 Graphical solution of equations

Exercise 29.1 (page 303) 1 a,b

x3

y = x3

y = x2 – 4

421–1–2–3–4 O

5

–5

–10

–15

–20

–25

–30

10

15

20

25

30y

c x = −1.3

2

x3

y = x2 – 2x

y = 9

4 521–1–2–3 O

2

–2

4

6

8

10

12y

x = 4.15 and x = −2.15


3 a x = 2 and x = −3b x = 3.8 and x = −0.8

4 a x = 2.65 and x = −2.65b x = 2.8 and x = −1.8c x = 0.4 and x = −2.4

5 a

x −3 −2 −1 −0.5 −0.25 0 0.25 0.5 1 2 3

y 0.11 0.25 1 4 16 – 16 4 1 0.25 0.11

x

y = 1x2

y = 2x + 4

2 31–1–2–3 O

2

4

6

8

10

12

14

16

18y

b y = 2x + 4c x = 0.45, x = −0.6 and x = −1.85


6 a

x

y = x2 – 4x

2 3 4 5 6 71–1–2–3 O

2

–2

–4

–6

4

6

8

10

12

14y

b i x = 0 and x = 4ii x = 5.15 and x = −1.15iiix = 1 and x = 3iv x = 2.6 and x = 0.4

7 a

x

y = x2 + 4x + 5

–2 –1 1 2 3–3–5 –4–6–7 O

2

4

6

8

10

12

14

16

18y


b i x = 0.3 and x = −3.3

ii x = −0.45 and x = −4.55

8 a y = 3b y = 2c y = −3xd y = x + 6

9 a y = 0b y = −1

2 c y = −6xd y = −41

2x

10 a x2 + 6x − 2 = 0b x3 − 4x − 1 = 0c x2 − 6x + 6 = 0d 2x2 − 4x = 0

11 a

x

y = 20x

4 5 6 7 8 9 10 1131 2O

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21y


b i x = 3.58ii x = 2.76 and x = 7.24iiix = 4.47

12 a

x

y = 2x

2 431–1–2–3–4 O

2

4

6

8

10

12

14

16

18y

b i x = 2.8ii x = 1 and x = 2iiix = 1

c 21.5 = 2.8


13 a

x −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

y 27 15.6 9 5.2 3 1.7 1 0.6 0.3 0.2 0.1

b

x

y = 3–x

1 2–1–2–3 O

2

4

6

8

10

12

14

16

18

20

22

24

26

28y

c i x = −1.46ii x = −1.89iiix = −2.77

14 a 1 solutionb 3 solutionsc 2 solutionsd 1 solution


Chapter 30 Inequalities


1 a trueb falsec trued falsee falsef trueg trueh truei true

2 a 5 or 4 or 3 …b −2 or −3 or −4 …c −4 or −3 or −2 …d 10 or 11 or 12 …e 20 or 19 or 18 …f −6 or −5 or −4 …g 5 or 6 or 7 …h −5 or −6 or −7 …i −2 or −1 or 0 …

3 a i 7, 8, 9, 10 and 11ii −2, −3, −4, −5 and −6iii−4, −3, −2, −1 and 0iv 0, −1, −2, −3 and −4

b i −3, −2, −1, 0, 1 and 2ii 0, 1, 2, 3, 4 and 5iii−5iv −2, −1, 0 and 1

4 a a < 12b a ≤ 4c a < 6d a ≥ 3e 4 < a < 8

5 a

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8

b

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8

c

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8

d

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8

e

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8

f

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8

6 a

–4 –3 –2 –1 0 1

b

–4 –3 –2 –1 0 21

c

–2 –1 0 1 2 43


d

1 2 3 4 5 76 8

e

–4 –3 –2 –1 0 21 3

f

–5 –4 –3 –2 –1 10

7 a x > 2b x ≤ 7c −1 < x < 3d −1 ≤ x < 3e x < 8f −3 < x ≤ 0

8 a 6b 4c 2d −2e 3

9 1 < x < 7 and 2 ≤ x ≤ 6 are two examples.

10 2 ≤ x < 4


1 x > 5

2 x < 4

3 x ≤ 12

4 t > 6

5 a ≤ 2

6 x < 2

7 n ≤ −134

8 p < 2

9 x < 3

10 x ≤ 114

11 x < −2

12 t < 1712

13 x ≥ 114

14 a > 1

15 m > −12

16 x > 10

17 k > −56

18 x < 2

19 a ≤ 214

20 x > −24

21 2 ≤ n < 4; integer solutions are 2 and 3

22 −1 < n < 1; integer solution is 0

23 −4 < n < 6; integer solutions are −3, −2, −1, 0, 1, 2, 3, 4 and 5

24 a ≤ −1 and a > 1

25 a ≤ 3 and a ≥ 4

26 width < 512 cm

27 x < 3.8

28 £23.99


1

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4

5x = 4

y

R

2

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4

5

y = –2

y

R

3

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4

R

5x = 0

y

4

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4

R

5x = 3x = –1

y

5

2 3 4 5 6 x1–1–2–3

1

–1O

–2

–3

2

3

4R

5

6x + y = 6

y


6

2 3 4 5 6 x1–1–2–3

1

–1O

–2

–3

–4

–5

–6

–7

2

3

4

R

2x – y = 6

y

7

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4R

5y = x

y

8

2 3 4 5 x1–1–2–3–4

1

–1O

–2

–3

–4

–5

2

3

4

R

5y = 3x + 1

y

9

2 3 4 5 6 7x1–1–2

1

–1O

–2

–3

–4

2

3

4

R

5

6

2x + 3y = 12

y

10

2 3 x1–1–2–3–4–5–6

1

–1O

–2

–3

–4

–5

2

3

4R

5

5y – 2x = 0

y

11

2 3 4 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3

4

R

5

3y – x = 9

y

12

2 3 4 5x1–1–2–3–4

1

–1O

–2

–3

–4

–5

2

3

4

R

5y = 2x

y

13

2 3 4 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3

4

R

5

x + 4y = –8

y


14

2 3 4 5 6 x1–1–2–3

1

–1O

–2

–3

–4

–5

2

3

4

R

5

y = 3x –� 4

y

15

2 3 4 5 6 x1–1–2–3

1

–1O

–2

–3

–4

–5

2

3

4

R

54x + 5y = 20

y


1

2 3 4 5 6 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3

4

R

5

6

7

3x + 2y = 12

x = 0

y = 0

y

2

2 3 4 5 6 7 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4

R

5

x = 6

y = 3

y

3

2 3 4 5 6 7 x1–1–2–3–4

1

–1O

–2

–3

–4

–5

2

3

4

R

5

6

7x + y = 6

x = 0

y = 0

y

4

2 3 4 5 6 x1–1–2–3–4–5

1

–1O

–2

–3

–4

2

3

4R

5

6

7

3x + y = 6

y = 2

x = 4

y


5

2 3 4 5 6 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3

4

R

5

6

y = 2x + 1

y = x + 1

x = 2

y

6

2 3 4 x1–1–2–3–4–5

1

–1O

–2

–3

–4

2

3

4

R

5

x + y = 2

x = –1

y

7

2 3 4 5 6 x1–1–2–3–4–5–6

1

–1O

–2

–3

–4

–5

2

3

4R

5

6

72x + y = 6

y = x

y = 4

y

8

2 3 4 5 6 x1–1–2–3–4–6 –5

1

–1O

–2

–3

–4

–5

2

3

4

R

5

y = x – 212

y = –1

x = –5

y

9

2 3 4 5 6 7 x1–1–2–3–4–5

1

–1O

–2

–3

–4

2

3

4

R

5

6

7

y = 1

y – 2x = 0

x + y = 5

y

10

2 3 4 5 x1–1–2–3–4–5

1

–1O

–2

–3

–4

2

3

4

R

5

x = 3

2x + 3y = 6

y = 3x

y


1 y ≤ 3 and x ≥ −1

2 x ≤ 3, y ≥ 0 and y ≤ x + 1

3 y < 4 and y ≤ 2x

4 y ≤ 2, y ≤ x, y > x − 2 and y + x ≥ −2

5 y ≥ 1, x ≥ −3 and y < −12x + 2


6 y ≥ 2, x ≤ 4, y ≥ −x + 3

and y ≤ 34x + 3


1 (5, 6), maximum 17

2 (2, 4), minimum 4

3 (4, 7), maximum 11

4 (3, 2), minimum 9

5 (6, 5), maximum 13

6 (2, 6), maximum 2


31 Angles in circles


5 b OMP = 90°

6 e angle POQ = 2 × angle PRQ

7 e all equal

8 d both angles 90°

9 c ii 180°d ii 180°

10 d 90°

11 e both equal


1 28°

2 162°

3 c = 18°, d = 70°, e = 70°

4 f = 55°, g = 45°, h = 45°

5 i = 96°, j = 264°, k = 132°

6 l = 58°, m = 39°, n = 83°


1 a = 116°, b = 65°

2 c = 127°, d = 106°

3 e = 48°, f = 118°

4 g = 65°, h = 65°, i = 115°

5 j = 46°, k = 92°

6 l = 29°, m = 151°


1 a = 36°, b = 108°, c = 54°

2 d = 46°, e = 67°, f = 23°

3 g = 26°, h = 52°, i = 26°

4 j = 122°, k = 61°

5 l = 74°, m = 106°

6 n = 108°, p = 36°, q = 54°, r = 54°


1 a = 74°

2 b = 35°

3 c = 38°, d = 38°

4 e = 52°, f = 26°

5 g = 77°, h = 154°

6 i = 46°, j = 29°, k = 105°


1 a 46°b 23°c 23°d 67°

2 a AED = 24°b CAD = 66°c DAB = 24°d ABC = 42°e ADB = 114°

3 a DCA = 28°b CAD = 76°c DAB = 14°d ADB = 104°e ABD = 62°

4 a CAD = 73°b ACD = 34°c AED = 17°d DAB = 17°e DBA = 56°

5 a CAD = 69°b ACD = 42°c AED = 21°d ADB = 111°e ABD = 48°

6 a ACB = 142°b CAB = 19°c BAT = 71°d ATB = 38°e CBD = 35.5°

7 a ADB = 58°b CAB = 32°c BAT = 58°d ATB = 64°e CBD = 29°

8 a BDE = 126°b ECB = 108°c CEB = 36°

9 a EAB = 48°b ECB = 96°c CEB = 42°

10 a ADB = 24°b ACB = 48°c CAB = 66°

11 a TAB = 32°b ACB = 64°c CAB = 58°

32 Polygons


1 a = 26°, b = 71°, c = 79°, d = 96°, e = 88°, f = 92°


2 a = 122°, b = 119°, c = 123°, d = 121°, e = 117°, f = 118°, g = 62°

3 a = 66°, b = 117°, c = 55°, d = 122°, e = 86°, f = 148°, g = 32°

4 a = 38°, b = 139°, c = 48°, d = 132°, e = 33°, f = 137°, g = 41°, h = 68°, i = 112°


1 a pentagonb octagonc nonagon

2 120°

3 135°

4 162°

5 165°

6 5 sides

7 10 sides

8 360 ÷ 14 is not a whole number

9 8 sides

10 The sum of the angles = 90 + 120 + 135 = 345° and not 360°


1 900°

2 3060°

3 13 sides

4 38 sides

5 82°

6 111°

7 23°

8 126°

9 54°, 108°, 108°, 108°, 162°

10 30°


1 a POQ = 72°b OPQ = 54°c QRS = 108°

2 a FOE = 60°b FEO = 60°c equilaterald CBA = 120°

3 12 sides

4 20 sides

5 24°

6 14.4°

7 a360n

b360n

c 180 − 360n

8 a 180 − xb x

33 Pythagoras and trigonometry


1 r 2 = p2 + q2

p2 = r 2 − q2

q2 = r 2 − p2

2 y2 = x2 + z2

x2 = y2 − z2

z2 = y2 − x2

3 m2 = l 2 + n2

l 2 = m2 − n2

n2 = m2 − l 2


1 2.5 cm

2 41 cm

3 19.5 cm


4 8.60 cm

5 8.54 cm

6 13.6 cm


1 15 cm

2 5.60 cm

3 10 cm

4 16.0 cm

5 2.95 cm

6 14.4 cm


1 7.75 cm

2 31.2 km

3 0.0862 m

4 9.85 cm

5 2.83 cm

6 5.59 cm


1 tan x = 54

2 cos x = 25

3 tan x = 47

4 cos x = 710

5 tan x = 25

6 sin x = 49

Exercise 33.6 (page 351) 1 4.50 cm

2 9.10 cm

3 19.3 cm

4 13.9 cm

5 3.24 cm

6 2.80 cm


1 4.45 cm

2 29.2 cm

3 2.44 cm

4 19.9 cm

5 4.92 cm

6 2.89 cm


1 63.4°

2 16.6°

3 36.9°

4 26.4°

5 29.7°

6 68.0°

7 2.38 cm

8 3.99 cm

9 74.1°

10 3.17 cm

11 10.9 cm

12 36.9°

13 2.29 cm

14 6.49 cm

15 19.5°

16 4.88 cm


1 a 14 kmb 126°

2 a 4.1 cmb 8.8 cm

3 a 8.90 cmb 13.8 cm

4 a 1200 mb 1500 m

5 a 120 mb 165 m

6 a 39°b 11 m


1 a 5.97 cmb 6.26 cmc 17.7°

2 a 25 cmb 23.7 cmc 69.2 cm

3 a 52.5°b 3.65 cmc 1.33 cm

4 a 10.6 cmb 21.2 cmc 40.3°d 13.5 cm

5 a 42.6 cmb 77.2 cmc 75.7 cm

6 a 19.8 cmb 37.9 cmc 31.8 cm

7 a 34.7 cmb 38.9°c 44.6 cm

8 a 15.8 cmb 22.9 cmc 20.9 cm


34 Similarity


1 corresponding angles are equal, ratios of sides = 1.4



4 corresponding angles are

equal, ratios of sides = 53




8 same shape – all radii the same so all corresponding measurements are in the same ratio

9 a corresponding angles are equal and all sides are equal in length

b the ratio of the corresponding sides can be different, e.g. a 7 by 5 rectangle is not similar to a 6 by 9 rectangle


1 x = 12 cm, y = 3.5 cm

2 x = 6.8 cm, y = 5.2 cm

3 x = 1.38 cm, y = 7.4 cm

4 x = 21.6 cm, y = 8.5 cm

5 x = 7.2 cm, y = 6.2 cm

6 x = 13 cm, y = 1.1 cm


1 corresponding angles A and D, B and F, E and C

2 PQXY

= PRXZ =

QRYZ = 1.25

3 corresponding angles C and J, B and K, D and H

4UWMN =

VWLN =

VULM = 1.15

5 corresponding angles ACB and DCE, BAC and DEC, ABC and CDE

6YZAC =

XZAC =

XYAB = 0.7


1 P = P (common angles), T = R (corresponding angles), S = Q (corresponding angles)

2 P = Y (alternate angles), Q = X (alternate angles), POQ = YOX (vertically opposite angles)

3 AOD = BOC (vertically opposite angles), DAO = OCB (alternate angles), ADO = OBC (alternate angles)

4 COD = FOE (vertically opposite angles), C = F (alternate angles), D = E (alternate angles)

5 ratios all 1.25

6 ratios all 2.125


1 a 2.8 cmb 4 cm

2 a 19 cmb 16.5 cm

3 a 12 cmb 1.68 cm

4 a 10 cmb 2.88 cm

5 a 9.6 cmb 7.2 cm

6 a 7 cmb 3.2 cm

7 a corresponding angles A and E, B and F, C and D

b i AC = 4.8 cmii DF = 7.84 cm

8 a corresponding angles G and L, H and J, I and K


b i GI = 5.8 cmii JL = 5.46 cm


1 a 2 : 5b 8 : 125

2 a 3 : 7b 27 : 343

3 a 2 : 3b 4 : 9

4 a 3 : 8b 27 : 512

5 a 3 : 2b 27 : 8

6 a 130.5 cm2

b 76 cm3

7 a 13.6 cmb 445.5 cm2

8 a 19 cmb 1120 cm3

9 a 86.4 cm2

b 4000 cm3

10 a 7.4 cmb 202.5 cm2

Chapter 35 Trig rules


1 14.3 cm

2 6.64 cm

3 4.77 cm

4 3.56 cm

5 7.16 cm

6 3.64 cm


1 43.4°

2 49.2°

3 73.6°

4 45.8°

5 94.2°

6 50.7°


1 6.38 cm

2 5.65 cm

3 5.04 cm

4 10.2 cm

5 7.56 cm

6 4.26 cm


1 58.8°

2 116.2°

3 60.1°

4 87.7°

5 60.7°

6 73.7°


1 5.47 cm2

2 89.4 cm2

3 23.9 cm2

4 16.6 cm2

5 14.0 cm2

6 31.5 cm2

7 8 cm

8 4.76 cm

9 56.4°

10 19.5°


1 a 3.49 cmb 14.3 cm2

2 a 86°b 8.33 cmc 18.5 cm2


3 a 74.3°b 45 cm2

4 a 10.8 kmb 171.2°

5 12.8 m

6 a 7.26 cmb 51.8 cm2

7 a 19.66 cmb 22.6 cm2

Chapter 36 Perimeter, area and volume 1


1 a 64 cmb 76 cm2

2 a 40 cmb 76 cm2

3 a 100 cmb 216 cm2

4 a 86 cmb 174 cm2

5 a 95 cmb 264 cm2

6 a 76 cmb 180 cm2


1 a 20πb 12πc 5.6π d 52π

e 535π

f 18π

2 a 15 cmb 28 cmc 6.8 cm

d 323 cm

3 a 18 cmb 4.5 cmc 4.2 cm

d 138 cm


1 a 56.5 cmb 107 cmc 19.5 cmd 9.8 cme 15.7 cmf 34.6 cmg 17.2 cmh 157 cmi 352 cm

2 a 10.8 cmb 3.06 cmc 8.91 cmd 15.9 cme 52.8 cm

3 a 7.63 cmb 0.40 cmc 1.34 cmd 5.57 cme 14.3 cm


1 11.8 cm

2 165 cm

3 29.8 cm

4 231 cm

5 2.86 cm

6 28.6 cm

7 33.6 cm

8 71.4 cm

9 a 36.9 cmb 33.3 cmc 46.8 cmd 59.4 cm


1 129 cm2

2 150 cm2

3 23.1 cm2


4 3.91 cm

5 1.38 cm

6 26.2 cm

7 a 1.69 cmb 10.6 cm

8 a 14.9 cmb 93.8 cm

9 0.716 cm2

10 682 cm2


1 190 cm2

2 6.28 cm2

3 2150 cm2

4 37.7 cm2

5 573 cm2

6 6360 cm2

7 22.9 cm2

8 1020 cm2

9 a 30.5 cm2

b 54.9 cm2

c 64.6 cm2

d 196 cm2


1 a 36 cm2

b 39.48 cm2

c 384 cm2

d 499.2 cm2

2 a 34.72 cm2

b 38.27 cm2

c 975 cm2

d 52.185 cm2

3 4.5 cm

4 13.6 cm

5 9.5 cm

6 3.36 cm


1 a 232 cm2

b 224 cm3

2 a 758 cm2

b 1386 cm3

3 a 162.78 cm2

b 133.38 cm3

4 a 4.7 cmb 124.6 cm2

5 a 6.4 cmb 390.4 cm2

6 a 14 cmb 1288 cm2

7 a 4.2 cmb 71.82 cm3

8 a 21 cmb 11 760 cm3

9 a 5 cmb 750 cm3

10 a = 72x2 and V = 36x3 so

A = 2Vx


1 19 800 cm3

2 294 cm3

3 18.2 cm

4 13 cm2

5 9.6 cm

6 88 cm2

7 a 1890 cm3

b 1368 cm2

8 a 3800 cm3

b 1779 cm2

9 a 7200 cm3

b 3840 cm2

10 a 23 040 cm3

b 5840 cm2


1 a 90 500 cm3

b 7540 cm2

c 9350 cm2

2 a 4630 cm3

b 965 cm2

c 1540 cm2

3 a 693 cm3

b 330 cm2

4 2800 cm3

5 92 800 cm3

6 1200 cm3

7 82.4 cm2

8 a 792 cm3

b 892 cm2

9 a 302 cm3

b 506 cm2

10 a 28 cmb 36 cm


37 Perimeter, area and volume 2


1 a 226 cm2

b 290 cm2

c 326 cm3

2 a 234 cm2

b 326 cm2

c 388 cm3

3 a 96 cm2

b 134 cm2

c 103 cm3

4 a 416 cm2

b 612 cm2

c 967 cm3

5 a 70.9 cm2

b 91.6 cm2

c 58.1 cm3

6 a 1630 cm2

b 2710 cm2

c 7530 cm3


1 3310 cm3

2 251 cm2

3 109 cm3

4 1960 cm2

5 44.8 cm2

6 86.9 cm3

7 A = 4πr 2 and V = 2 2πr 3

3

So VA =

2 2πr 3

3 ÷ 4πr 2

= 2r 6


1 545 cm3

2 240 cm3

3 462 cm3

4 139.5 cm3

5 484.5 cm3

6 10.6 cm3

7 252 cm3

8 1054 cm3

9 1332 cm3

10 1947 cm3


1 a 3630 cm2

b 20 600 cm3

2 a 266 cm2

b 408 cm3

3 a 5540 cm2

b 38 800 cm3

4 a 33 900 cm2

b 452 000 cm3

5 a 339 cm2

b 452 cm3

6 72.8 cm3

7 70 cm2

8 40.3 cm2

9 205 cm3

10 85.4 cm2


1 a 207 cm3

b 210 cm2

2 a 23.2 cm3

b 59.3 cm2

3 45 spheres

4 148 cm3

5 a 50.3 cm3

b 68.6 cm2

6 87 cm3

7 1810 cm3


1 a 10.1 cmb 45.2 cm2

2 a 15.7 cmb 56.5 cm2

3 a 3.13 cmb 10 cm2

4 a 57.2 cmb 572 cm2

5 a 31.8 cmb 556 cm2

6 a 17.1 cmb 120 cm2


1 a 49.1°b 10.3 cm2

2 a 6.42 cmb 8.40 cm

3 a 8.56 cmb 55.6 cm2

4 a 20.1°b 25.3 cm2


5 a 75.5°b 8.43 cm

6 240 cm2

7 4.72 cm

8 9.21 cm2

9 16.5 cm

10πxt180 :

πxt 2

360 = 2 : t


1 9.97 cm, 1.30 cm2

2 7.43 cm, 0.958 cm2

3 8.35 cm, 0.607 cm2

4 7.01 cm, 1.47 cm2

5 0.778 cm, 14.1 cm2

6 0.0894 cm, 2.66 cm2


1 length

2 area

3 area

4 volume

5 area

6 length

7 volume

8 area

9 volume

10 volume


1 length

2 area

3 volume

4 volume

5 none of these

6 length

7 area

8 none of these

9 area

10 none of these

11 length

12 none of these

13 volume

14 area

38 Constructions, loci, plans and elevations


All other answers are constructions.

9 b i 72.4 kmii 037


These are all constructions; no answers supplied.


These are all constructions; no answers supplied.



1

a b c

2

a b c

3

a b c

4

a b c

5

a b c

6

a b c


Chapter 39 Compound measures


1 301 miles

2 47 km/h

3 5 hours

4 8 hours

5 62 km/h

6 8 hours

7 8 hours

8 216 miles

9 7 hours

10 53 km/h


1 119 miles

2 30 km/h

3 4 h 15 min

4 210 miles

5 3 h 40 min

6 7 h 32 min

7 65 km/h

8 198 miles

9 45 km/h

10 355 miles


1 56 mph

2 39 miles

3 15.25

4 152 miles

5 55 mph

6 10.55

7 55 miles

8 65 mph

9 22.37

10 76 miles


1 38 mph

2 26 mph


3 23 mph

4 48 mph

5 42 mph

6 40 mph

7 12 mph

8 15 mph

9 36 mph

10 42.8 mph


1 18 km/h

2 15 m/s

3 72 km/h

4 4 m/s

5 28.8 km/h

6 7 m/s

7 162 km/h

8 60 m/s

9 10.8 km/h

10 35 m/s


1 5.6 g/cm3

2 34.2 g

3 16 cm3

4 8.4 g/cm3

5 181.7 g

6 16.5 g/cm3

7 6 cm

8 2704 g

9 54 cm3

10 14 g/cm3


1 £1785

2 128 people/km2

3 £27.56/m2

4 56 250 000 people

5 2.5 kg

6 147 ml of concentrate

7 24.5 people/km2

8 £9.48 per book

9 3.4 m

10 180 000 km2

Chapter 40 Bounds


1 47.5 kg, 48.5 kg

2 5.25 cm, 5.35 cm

3 4.275 ml, 4.285 ml

4 35 s, 45 s

5 4500 mm, 5500 mm

6 57.5 cm2, 58.5 cm2

7 395 ml, 405 ml

8 4950 mg, 5050 mg

9 58.5 ml, 59.5 ml

10 5.9995 km, 6.0005 km


1 a 21.8 cmb 30.3125 cm2

2 5.85 to 5.92 cm

3 a 29.8 cmb 73.9 cm2

4 13 125 to 39 375 cm3

5 50.9 to 56.4 mph

6 4.78 to 4.97 g/cm3

7 4.05 to 5.26 cm

8 a 9.32 cmb 20.6 cm2

9 a 6.75 cmb 1596 cm3

10 3.43 to 3.69 cm2


1 a i 128 ii 1.42

b i 3858.75ii 20

2 a i 23.681 525ii 0.59

b i 9.73ii 1.12

3 a i 13.9ii 2.33

b i 39.6325ii 3.09iii5.3

4 a i 81 250ii 15.8iii175


b i 65ii 22 500iii1.26iv 0.462

5 a i 8.5ii 3.5775iii1.8225iv 2.29v 1.4vi6.8vii0.243

b i 7.1875ii 3.8iii1.60iv 4.4v 0.472vi10.9vii2.30

Chapter 41 Transformations


1

2 3 4 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4y B

A C

A'

B'

C'

2

2 3 4 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4y

F

E

D

F'

D'

E'


3

2 3 4 5 6 7 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3

4

5y

G

H

I

x = 1

G'

I'

H'

4

2 3

y = 2

4 x1–1–2–3–4

1

–1O

–2

–3

–4

2

3

4

5

6

7

8y

J

L

K

K'

L'

J'

5

2 3 4 5 x1–1–2–3–4–5–6–7–8–9

1

O

2

3

4

5

6x = –2

y

P

M

N

P'

M'

N'

6

2 3 4 x1–1–2–3–4

1

–1O

–2

–3

–4

–5

–6

–7

–8

–9

–10

2

3

4y

S

R

Q

F'

Q'

S'

R'

y = –3

7

2 3 4 5 6 7 x1

1

O

2

3

4

5

6

7y

T

U

V

y = x

T'

V'

U'

8

2 3 4 5 6 x1–1–2–3–4–5–6

1

–1O

–2

–3

–4

–5

–6

2

3

4

5

6y

Z

YX

y = –x

X'

Y'Z'



1

2 3 4 5 x1

1

–1

O

–2

–3

–4

–5

2

3

4

5y

B

A

C B'

C'

A'

2

2 3 4 5 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

y

A

B

C x = –1

B'

A'

C'

3

2 3 4 5 6 x1–1–2–3–4–5–6

1

–1O

–2

–3

–4

–5

–6

2

3

4

5

6y

A

C

B

y = x

B'

A'

C'

4

2 3 4 5 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

y

A

BC

B'

A'C'

5

2 3 4 5 x1

1

–1

O

2

3

4

5

6

7y

B

A

C

y = 3

C'

B'

A'

6

2

y = –x

3 4 5 6 x1–1

O

–2

–3

–4

–5

–6

y

A

C

BC'

A'

B'


1 reflection in y = x

2 reflection in x-axis (y = 0)

3 reflection in y = −1

4 reflection in y = −x


1

2 3 4 5 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3

4

5y

CA

B

A'

B'

C'


2

2 3 x1–1–2–3

1

–1O

–2

–3

2

3y

F

D

E

E'

F'

D'

3

2 3 4 5 6 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2

3y

G

I

H

H'

I'

G'

4

2 3 4 x1–1–2–3–4

1

–1O

–2

–3

–4

–5

–6

2

3

4

5

6

7

8

9y

L

J

K

L'

K'

J'

5

2 3 4 5 x1–2 –1–3–4–5–6

1

O

2

3

4

5

6y

P

N

M

M'

N'

P'

6

2 3 4 5 x1–2 –1–3–4

1

O

2

3

4

5

6

7y

S

R

Q

R'S'

Q'

7

2 3 4 x1–2 –1–3–4–5–6

1

O

2

3

4

5

6

7y

V

U

T

V'

T'

U'

8

x–1–2–3–4–5–6

1

–1O

–2

–3

–4

–5

2

3

4y

X

Y

Z

X'

Z'

Y'


1

2 3 4 5 x1–1–2–3–4–5

1

–1O

2

3

4

5y

C

A

B

A'

B'

C'


2

2 3 4 5 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2y

C

B

A

A'

B'

C'

3

x–1–2–3–4–5–6

1

–1O

2

3

4

5y

C

A

B

A'

B'

C'

4

2 x1–1–2–3–4–5

1

–1O

–2

–3

–4

–5

2y

C

A

B

C'

A'

B'

5

2 3 4 5 6 7 8 x1

1

–1

O

–2

–3

–4

–5

–6

2

3

4

5

6

7y

A

C

B

A'

C'

B'

6

2 3 4 5 6 x1

1

–1

O

–2

–3

–4

–5

2

3

4

5

6y C

A

B

B'

C'

A'


1 90° clockwise rotation about (0, 1)

2 90° clockwise rotation about (1, 1)

3 90° anti-clockwise rotation about O

4 90° anti-clockwise rotation about (−1, −1)


1

2 3 4 5 6 x1

1

O

2

3

4y B

CA

B'

C'A'

2

2 x1–1–2–3–4–5

1

–1O

2

3

4

5

6

7

8y

F

D

E

D'

E'

F'


3

x1–1–2–3–4–5

1

–1

–2

–3

–4

O

y

H

G

I

G'

H'

I'

4

2 3 4 x1–1

O

–2

–3

–4

–5

–6

–7

y

K'

L'

J'K

L

J

5

2 3 4 5 x1

1

O

2

3

4

5

6y

P

N

M

P'

N'

M'

6

x1–1–2

1

–1O

–2

–3

2

3

4y

S

Q

R

S'

Q'

R'

7

2 3 4 x1

1

O

2

3

4

5

6

7

8

9

10y

V

U

TV'

U'

T'

8

x–1–2–3–4–5–6–1O

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

X

Y

Z

X'

Y'

Z'


1

2 3 4 x1–1

1

–1O

2

3

4

5

6y

C

A

B

C'

A'

B'

2

x–1 1 2 3–2–3–1O

–2

–3

–4

–5

–6

–7

y

C

B

A

C'

B'

A'

3

x–2 –1 1–3–4

6

7

8

9

10

O

5

4

3

2

1

yA

B

C

A'

B'

C'

4

x–1 1–2–3–1O

–2

–3

–4

–5

–6

–7

y

B

A

C

B'

A'

C'

5

x–2 –1 1 2 3 4 5

6

7

8

9

10

O

5

4

3

2

1

yA

B

C

A'

B'

C'

6

2 3 4 5 6 7 8 x1–1

–2

–3

–4

–5

–6

–7

–8

–9

O

y

A

B

C

A'

B'

C'



1 translation (43) 2 translation ( –1

–2) 3 translation (30) 4 translation ( 0

–2)Exercise 41.10 (page 462)

1

2 3 4 5 6 7 8 x1

8

7

6

5

4

3

2

1

O

y

A

B

C

A'

C'

B'

2

2 3 x1–1–2–3–4–5–6

1

–1O

–2

–3

2

3

4y

F

E

D

F'

D'

E'

3

2 3 4 5 6 7 8 3 10 x1–1–2–3–4–5–6

1

–1O

–2

–3

2

3

4y

I

H

G I'

G'

H'

4

2 3 4 5 6 7 8 9 10 x1–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

–13

O

y

K

J

L

K'

J'

L'

5

2 3 4 5 6 x1

6

5

4

3

2

1

O

y

M

N

PM'

N'

P'

6

x–2 –1 1 2–3–4

6

7

8

9

O

5

4

3

2

1

y

R

Q

S

R'

Q'

S'


7

2 3 4 x1–1–2–3

1

–1O

–2

–3

–4

–5

2

3

4

5

6

7y

V

T

U

T'

U'

V'

8

x–1–2–3–4–5–6–1O

–2

–3

–4

–5

–6

–7

–8

y

X

Y

ZX'

Y'

Z'


1

2 3 4 x1

1

–1O

2

3

4

5y

A

C

B

C'

A'

B'

2

2 3 4 5 x1

1

–1

O

–2

–3

–4

–5

–6

–7

y C

A

A

B'

C'

A'

3

2 x1–1–2–3–4

1

–1

–2

O

2

3

4

5y

A

B

C

A'

B'

C'

4

x–1 1–2–3–4–5–6–7–8–9–1O

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

–13

–14

–15

–16

–17

y

B

A

C

B'

A'

C'

5

2 3 4 5 x1

1

O

2

3

4

5

6

7y

C

A

B

C'

A'

B'

6

2 3 4 5 6 x1–1

–2

–3

–4

–5

Oy

C

A

B

C'

A'

B'



1 enlargement, centre O, scale factor 212

2 enlargement, centre (1, 0), scale factor 2

3 enlargement, centre O, scale factor −2

4 enlargement, centre (−1, 2), scale factor 12


6 enlargement, centre O, scale factor 112


1

2 3 4 x1–1

1

–1O

–2

–3

2

3

4y

CA

B

A'

B'

C'

2

2 x1–1–2–3–4

1

–1O

–2

2

3y

F

D

E

F'

E'

D'

3

x–1–2–3–4–5–6–7–8–9–10–11–12–1

1

2

3

4

5

6

O

–2

y

H

G

I

H'

G'

I'

4

2 3 4 x1–1–1O

–2

–3

–4

y

K

LJ

J'

L'

K'

5

2 3 4 5 6 7 8 9 10 11 12 x1

1

O

2

3

4

5

6

7y

M

P

N

M'

P'

N'

6

2 x1–1–2

1

–1O

–2

–3

–4

–5

–6

–7

–8

–9

2

3

4y

Q

R

S

Q'

S'

R'



2 translation ( 1–3)

3 180° rotation about (−1, 1)

4 reflection in x = −2

5 90° anti-clockwise rotation about (−2, 1)

6 reflection in y = x


Chapter 42 Questionnaires


1–6 Suitable suggestions.




1 a not balanced

2 a wrongly defined

3 a not closed

4 a gaps

5 a gaps

6 a wrongly defined

7 a not balanced

8 a gaps

9 a not closed

10 a wrongly defined






Chapter 43 Statistical diagrams


1

Numberofbrothersorsisters

Frequency

0 13

1 16

2 13

3 9

4 6

5 3

2

Numberofgoalsscored

Frequency

0 4

1 17

2 26

3 28

4 15

5 3

6 2

7 3

8 2

3

Breadthsofobjects(cm)

Frequency

1–5 3

6–10 8

11–15 12

16–20 11

21–25 11

26–30 5

4

Massofobjects,M(kg)

Frequency

5 < M ≤ 7 8

7 < M ≤ 9 13

9 < M ≤ 11 11

11 < M ≤ 13 14

13 < M ≤ 15 10

15 < M ≤ 17 2

17 < M ≤ 19 2

5

Heightsh(cm) Frequency

1–5 10

6–10 9

11–15 11

16–20 16

21–25 14

6

CapacityV(cm3) Frequency

12–19 12

20–27 8

28–35 10

36–42 12

43–50 6


1 a

Time(hours) Frequency

2 135

3 225

4 190

5 90

6 110

7 40

8 10

b range = 6c mode = 3d 31.25%e 27 : 22

2 a

Points Frequency

15 12

30 46

45 42

60 74

75 28

90 42

105 6

b range = 90c mode = 60

d 25

e 14 played the game again


3 aVolume,V Frequency

0 < V ≤ 6 1200

6 < V ≤ 12 1900

12 < V ≤ 18 600

18 < V ≤ 24 1650

b The limits of the modal class are 6 and 12.

c 45107

d 57.9%e 12 : 37

4 a Area,A Frequency

0 < A ≤ 5 28

5 < A ≤ 10 56

10 < A ≤ 15 18

15 < A ≤ 20 96

20 < A ≤ 25 74

25 < A ≤ 30 36

b The limits of the modal class are 15 and 20.

c 198

d 2477

e6177

Exercise 43.3 (page 489)Note that for questions 1 to 4 part b there may be alternative answers to those given here.

1 a

60 6550

Science

Maths

Mark, m

Freq

uenc

y

55454035

2

0

4

6

8

10

b The maths results are generally better than the science results / the maths results are more spread out than the science results.

2 a

7060

Number of goals, G

Freq

uenc

y

65555045

2

0

4

6

8

1

3

5

72010

2009

b There is more variation in the number of goal scored in 2009 than 2010.

3 a

28 3024Temperature, T (°C)

Freq

uenc

y

26222018

2

0

4

6

8

10

12

14

Athens

Belfast

b Athens is generally warmer than Belfast / there is more variation in the temperature in Belfast.


4 a

10 126

Length, l (letters)

Freq

uenc

y

8420

10

0

20

30

40

teenager’s book

child’s book

b The length of the words in the child’s book is generally shorter than those in the teenager’s book.

5 Mass,M(kg)

2<M≤4 4<M≤6 6<M≤8 8<M≤10 10<M≤12

cubes 2 6 8 4

spheres 3 12 5

6 Volume,V(cm3)

5<V≤10 10<V≤15 15<V≤20 20<V≤25 25<V≤30

copper 180 730 490

silver 110 370 680 240


1 3 1 2 2 3 5

4 2 4 5 6 6 7

5 0 1 4 4 5

Key: 3 |1 = 31 °F

2 2 1 3 5 6 6

3 0 4 6 7 8

4 2 5 6 7

5 1 4 4 9 9 9

Key: 2 |1 = 2.1 cm

3 2 2 3 5 5 6 7

3 4 4 5 6 7 8 9

4 1 2 3 5 6

5 1 1 4 6 8 9

Key: 2 |2 = 2.2 km


4 3 1 1 2 7 7 9

4 2 3 5 6 6 8 8

5 1 4 6 7 9

6 2 3 5 5 6 7 8 9

Key: 3 | 1 = 31 litres

5 girls boys

4 3 2 1 4 1 2 5 6

6 5 2 1 1 5 0 1 1 3 5

7 7 4 3 1 0 6 1 4 5 6 8

9 8 6 6 2 7 2 3 5 7 8 9

Key: 1 | 4 = 41 marks (girls) 4 | 1 = 41 marks (boys)

6 height length

4 2 2 1 4 1 3 5 6 7

6 5 4 3 1 5 2 4 6 7 8

7 7 6 5 4 2 6 1 3 5 8

9 9 8 7 3 7 1 1 4 6 7 9

Key: 1 | 4 = 4.1 cm (height) 4 | 1 = 4.1 cm (length)

7 women men

4 2 1 0 5 1 2 3 5 6

5 3 2 6 1 1 4 5 8

5 4 2 1 7 2 3 5 6

9 8 7 6 6 5 2 8 1 4 5 8

Key: 0 | 5 = £50 (amount spent by women)

5 | 1 = £51 (amount spent by men)

8flat objects round objects

8 7 5 4 3 1 2 1 3 5 6

9 6 4 2 1 1 3 1 2 4 5 6

5 4 3 3 2 1 4 1 3 5 7

3 1 5 1 1 2 7 8 8 9

Key: 1 | 2 = 2.1 kg (flat objects)

2 | 1 = 2.1 kg (round objects)


1 a range = 38 mmb mode = 35 mmc median = 33 mm

2 a range = 34 marksb mode = 53 marksc median = 59 marks

3 a range = 23 litresb mode = 93 litresc median = 86.5 litres

4 a range = 36 kgb mode = 24 kgc median = 31 kg

5 boys:a range = 46b mode = 74c median = 74

girls:a range = 45b mode = 72c median = 74

6 lengths:a range = 3.6 cmb mode = 5.1 cmc median = 5.1 cm

widths:a range = 3.6 cmb mode = 3.1 cmc median = 5.1 cm

7 mass:a range = 35 gb mode = 64 gc median = 64 g

volume:a range = 38 cm3 b mode = 42 cm3 c median = 61 cm3

8 female:a range = £34b mode = £62c median = £61

male:a range = £36b mode = £50c median = £62



1 negative correlation

2 zero correlation

3 positive correlation

4 zero correlation

5 negative correlation

6 positive correlation


1 b positive correlationc as the distance increases, the time taken

to travel increases

2 b positive correlationc as the number of books ordered

increases, the cost increases

3 b negative correlationc as the density increases, the volume

decreases

4 b negative correlationc as the Literacy mark goes up, the

Numeracy marks falls

5 b positive correlationc the larger the number in the group, the

greater the amount of money collected

6 b negative correlationc as the age of the car increases, its value

falls


1 43

2 7

3 448

4 2.5

5 950

6 21.21


Chapter 44 Statistical averages and spread


1 a range = 5b mode = 10c median = 9d mean = 9.4

2 a range = 25b mode = 10c median = 15d mean = 17


4 a range = 2.5b mode = 1.5c median = 1.75d mean = 1.95


6 a range = 1b mode = 1.2c median = 1.4d mean = 1.46


8 a range = 50b mode = 30c median = 45d mean = 45


10 a range = 1.2b mode = 6.2c median = 6.4d mean = 6.52


1 a i modal class 16 < l ≤ 20ii median class 12 < l ≤ 16

b mean = 11.6 mm

2 a i modal class 32 < h ≤ 40ii median class 16 < h ≤ 24

b mean = 20.6 cm

3 a i modal class 10 < P ≤ 15ii median class 15 < P ≤ 20

b mean = £20

4 a i modal class 12 < M ≤ 13ii median class 13 < M ≤ 14

b mean = 13.2 mg

5 a i modal class 0 < A ≤ 25ii median class 25 < A ≤ 50

b mean = 43.5 cm2

6 a i modal class 0 < D ≤ 6ii median class 12 < D ≤ 18

b mean = $13.5


7 a i modal class 38 < V ≤ 52

ii median class 38 < V ≤ 52

b mean = 41.5 cm3

8 a i modal class 31–40ii median class 21–30

b mean = 24.375 years


b mean = 12.75 mm


b mean = 7.16 absences


1 n = 4

2 n = 4

3 n = 5

4 n = 8

5 n = 21

6 n = 9

7 n = 2

8 n = 9


1 7.76 m

2 £6.96

3 5.204 litres

4 32.2 years

5 8.176 kg

6 £28

7 7.2 cm

8 58 marks


1 boys: range 4, mode 31, median 32, mean 31.95

girls: range 2, mode 31, median 31, mean 31.33… (31.3

. )

2 men: range 2, mode 7, median 6, mean 6.25

women: range 3, mode 4, median 8, mean 7.8


girls: range 3, mode 0, median 0, mean 0.52


girls: range 2, mode 16, median 16, mean 16.04

5 British: range 5, mode 5, median 4, mean 3.65

Foreign: range 4, mode 2, median 3, mean 2.86

6 bus: range 4, mode 3, median 2, mean 1.9

train: range 3, mode 2, median 2, mean 1.7


1 mode

2 mean

3 median

4 mode

5 mean

6 mode

7 mean

8 mean

9 mode

10 median


Chapter 45 Cumulative frequency curves and box plots


1 aHeight≤h(cm)

0 4 8 12 16 20 24 28

Cumulativefrequency

0 4 20 48 138 162 174 180

b

16 20 24 28Height, h (cm)

Cum

ulat

ive

freq

uenc

y

1284

Q1

Q2

Q3

0

20

0

40

60

80

100

120

140

160

180

c i 142.5 cm ii 4 cm

d 165

2 a

Mark(≤) 0 10 20 30 40 50 60

Cumulativefrequency

0 2 7 28 53 57 60

b

30 40 50 60Mark

Cum

ulat

ive

freq

uenc

y

2010

Q1

Q2

0

10

0

20

30

40

50

60


c i 31 ii 12

d 35

3 a Age(≤)(years) 10 16 22 28 34 40 46 52

Cumulativefrequency 0 8 32 70 158 200 228 240

b

40 50

Age (years)

Cum

ulat

ive

freq

uenc

y

3020

Q1

Q2

10

20

0

40

60

80

100

120

140

160

180

200

220

240

Q3

c i 32.5 years ii 9.5 years

d 26%

4 a Mass≤M(kg) 10 12 14 16 18 20 22 24 26

Cumulativefrequency 0 2 9 21 59 96 111 116 120


b

18 20 22 24 26

Mass (kg)

Cum

ulat

ive

freq

uenc

y

161412

Q2

Q3

10

20

0

40

60

80

100

120

Q1

c i 18 kg ii 2.6 kg

d 32

5 a Points(≤) 0 50 100 150 200 250 300 350 400


b

200 250 300 350 400Points

Cum

ulat

ive

freq

uenc

y

15010050

Q2

10

10

0

20

30

40

50

60

Q1

Q3

c i 198 points ii 108 points

d 10

6 a Width≤w(cm) 0 8 16 24 32 40 48 56 64 72

Cumulativefrequency 0 8 20 43 81 145 179 204 214 220


b

32 40 48 56 64 72Width, w (cm)

Cum

ulat

ive

freq

uenc

y

24168

Q2

0

40

0

80

120

160

200

20

60

100

140

180

220

Q1

Q3

c i 35.5 cm ii 17.5 cm

d 49

7 a Miles(≤) 0.5 15.5 30.5 45.5 60.5 75.5 90.5 105.5 120.5


b

60 75 90 105 120

Distance (miles)

Cum

ulat

ive

freq

uenc

y

453015

Q2

0

20

0

40

60

80

100

10

30

50

70

90

110

120

130

140

150

160

Q1

Q3

c i 61.5 miles ii 22 miles

d 4%

8 a Area≤A(cm2) 40 44 48 52 56 60 64 68 72



b

Area, A (cm2)

Cum

ulat

ive

freq

uenc

y

52 56 60 64 68 724844

Q2

40

20

0

40

60

80

100

10

30

50

70

90

110

120

130

140

150

160

170

180

190

200

Q1

Q3

c i 56.25 cm²

ii 4.75 cm²d 11

9 a Length≤l(cm) 0 20 40 60 80 100 120 140 160 180

Cumulativefrequency 0 2 7 17 39 79 104 113 120 124

b

Length, l (cm)

Cum

ulat

ive

freq

uenc

y

20 40 60 80 100 120 140 160 180

Q1

Q2

Q3

0

20

0

40

60

80

100

120


c i 91 cm ii 37 cm

d 103

10 a Cost≤C(£) 80 90 100 110 120 130 140 150 160


b

Cost, C (£)

Cum

ulat

ive

freq

uenc

y

90 100 110 120 130 140 150 160

Q1

Q2

Q3

80

40

0

80

120

160

200

240

260

280

20

60

100

140

180

220

c i £121 ii £27

d 166


1 a Volume,V(litres) Frequency

0 < V ≤ 16 4

16 < V ≤ 32 8

32 < V ≤ 48 24

48 < V ≤ 64 36

60 < V ≤ 80 28

72 < V ≤ 96 12

84 < V ≤ 112 8

b median 58 litres, IQR 28 litres


2 aHeight,h(cm) Frequency

130 < h ≤ 135 40

135 < h ≤ 140 80

140 < h ≤ 145 170

145 < h ≤ 150 190

150 < h ≤ 155 100

155 < h ≤ 160 60

b median 145.5 cm, IQR 8.5 cm

3 aNumber Frequency

1–20 10

21–40 25

41–60 50

61–80 40

81–100 30

101–120 5

b median 58 downloads, IQR 36 downloads

4 aCosts(£) Frequency

Over 200 and up to and including 240 2







b median £338, IQR £64

5 a

Age(years) Frequency

5–6 50

7–8 200

9–10 600

11–12 250

13–14 100

b median 9.2 years, IQR 2 years

6 aMass,Mg Frequency

10 < M ≤ 14 10

14 < M ≤ 18 25

18 < M ≤ 22 35

22 < M ≤ 26 60

26 < M ≤ 30 40

30 < M ≤ 34 35

34 < M ≤ 38 5

b median 24.5 kg, IQR 8.5 kg


1

38 40 42 44

Cost (£)46 48 50 52 5436

2

30 35 40 45

Area (cm2)50 55 60 6525

3

42 44 46 48

Width (mm)50 52 54 56 57 58 6040

4

25 30 35 40

Age (years)45 5020

5

130135140145150155160165170125

6

65 70 75 80 85 9060


Exercise 45.4 (page 530)Note that alternative answers are possible for questions 1to6.

1 The Year 12 marks are more spread out (a bigger range).

Year 8 students have generally done better (higher minimum mark, median and quartiles).

2 The number of hours spent revising by girls is more spread out (a bigger range).

The middle 50% of the girls’ hours are more spread out (a bigger box).

3 Teenagers own more CDs than adults (min, median, quartiles and max values all higher).

The number of CDs owned by teenagers is more spread out (a bigger range).

4 The number of mistakes made by women is more spread out (a bigger range).

In general women make fewer mistakes (lower median and quartile values).

5 The middle 50% of the men’s wages are more spread out (a bigger box).

The range of wages is the same for women and men but lower min and max values for women.

6 The middle 50% of the science and maths marks cover the same range.

More students have done better in maths (higher quartile and median values).


1 aMark(<) Cumulative

frequency

0.5 0

10.5 5

20.5 14

30.5 40

40.5 75

50.5 97

60.5 104

70.5 108

b

40 50 60 70 80

Marks

Cum

ulat

ive

freq

uenc

y

302010

Q1

Q2

Q3

0

20

0

40

60

80

100

c i 33 ii 26 iii43

d

10 20 30 40Marks

50 60 700

2 a Length≤l(cm) Cumulativefrequency

0 0

6 14

12 50

18 122

24 180

30 222

36 240

b

Length, l (cm)

Cum

ulat

ive

freq

uenc

y

12 18 24 30 366

Q1

Q2

0

40

0

80

120

160

200

220

240

20

60

100

140

180


c i 18 ii 13 iii24

d

10 20 30 40

Length (cm)0

3 a Age(≤) Cumulativefrequency

15 0

18 6

21 20

24 52

27 106

30 144

33 156

36 164

b

Age (years)

Cum

ulat

ive

freq

uenc

y

21 24 27 30 33 3618

Q1

Q2

Q3

15

20

0

40

60

80

100

120

140

160

c i 26 ii 23 iii36

d

20 25 30 35 40

Age15

4 a Mass≤Mg Cumulativefrequency

0 0

8 12

16 40

24 82

32 119

40 144

48 160

b

Mass, M (kg)

Cum

ulat

ive

freq

uenc

y

16 24 32 40 488

Q1

Q2

Q3

0

20

0

40

60

80

100

120

140

160

c i 23.75 ii 16 iii32.5

d

10 20 30 40 50Mass

0


5 a Cost(<)(£) Cumulativefrequency

60.50 0

80.50 5

100.50 17

120.50 42

140.50 69

160.50 80

180.50 88

b

Cost (£)

Cum

ulat

ive

freq

uenc

y

100 120 140 160 18080

Q1

Q2

Q3

60

20

0

40

60

80

c i £122 ii £106 iii£138

d

80 100 120 140 160 180

Cost60

6 a Lifetime≤L(hours)

Cumulativefrequency

9 0

10 3

11 12

12 36

13 74

14 122

15 143

16 151

17 156

b

Lifetime, L (hours)

Cum

ulat

ive

freq

uenc

y

13 15 1711 12 14 1610

Q1

Q2

Q3

9

20

0

40

60

80

100

120

140

160

c i 13.1 hours ii 12.1 hours iii13.9 hours

d

10 11 12 13 14 15 16 17Lifetime (hour)

9


Chapter 46 Histograms and sampling


1

100 11080Mass (g)

Freq

uenc

y de

nsity

907060

0.2

0

0.4

0.6

0.8

1.0

1.2


2

20 2510

Cost (£)

Freq

uenc

y de

nsity

1550

1

0

2

3

4

5

6

30 35

3

20 2510

Height (cm)

Freq

uenc

y de

nsity

1550

10

0

20

30

40

50

60

30 35

4

50Perimeter (cm)

5540

Freq

uenc

y de

nsity

4535300

10

20

30

60 65


5

20 2510

Scores

Freq

uenc

y de

nsity

1550

1

0

2

3

4

5

30 35

6

50

Time (minutes)

5540

Freq

uenc

y de

nsity

4535300

20

40

60

60 65


1 Cost,C(£) Frequency

1–6 21

7–11 28

12–15 10

16–25 48

26–30 8

31–35 16

2

Height,h(cm) Frequency

0 < h ≤ 3 162

3 < h ≤ 8 120

8 < h ≤ 18 440

18 < h ≤ 33 510

33 < h ≤ 35 88


3Price,P(£) Frequency

1–4 6

5–9 14

10–19 6

20–21 5

22–25 8

26–30 4

4Time,t(minutes) Frequency

30 < t ≤ 32 8

32 < t ≤ 35 6

35 < t ≤ 40 30

40 < t ≤ 50 120

50 < t ≤ 60 580

60 < t ≤ 65 260


1 a random b no, doesn’t include

people who don’t use the train, perhaps because they find it unreliable

2 a random b possibly, but sample

size is small and it is unclear what range of subjects is covered

3 a systematic b yes

4 a random b yes

5 a random b no, depends who those

100 people are and how they are selected

6 a random b no, needs to be done at

different times

7 Biased because boys are not asked.

8 Biased because only people who go to church are asked.

9 Biased because no young people are asked.

10 Biased because only students in the Additional Maths class are asked.


1 red 4, yellow 12, blue 18, green 16

2 Ford 9, Renault 15, Volkswagen 18, Skoda 6

3 24 males

4 Spain 6, Italy 18, France 22, Portugal 4

5 chicken 10, turkey 16, beef 14, vegetarian 8

6 football 16, rugby 20, netball 8, hockey 12

7 14 from full or half board

8 tennis 9, motor racing 12, ice hockey 24, hurling 18

9 train 6, car 10, walk 12, bus16

10 3


1 a 97 red b 194 bluec 388 green

2 a 194 menb 582 boysc 679 girls

3 a The Times 436b The Mirror 545

4 a Mini 363b Fiesta 484c Audi 605

5 a gym 146b badminton 438

6 a Irish 119b French 595

7 a 162 walkb 243 come by trainc 486 drive

8 a 492 visit friendsb 984 stay at home

9 a 134 men b 469 boys c 536 girls

10 a Australia 87b America 348c Egypt 435


1 a 3 menb 12 children

2 a 10 motor bikesb 25 cars

3 a 8 cubesb 28 cylinders

4 a 28 in Years 8 and 9 b 21 in Years 10 and 11

5 a 12 marblesb 16 countersc 20 beads

6 a 12 carpb 6 perchc 15 pike

7 15 Linfield, 40 Cliftonville, 35 Glenavon

8 a 8 menb 20 children


Chapter 47 Probability


1 a 925

b 1625

c 115

2 a P 0.24, Q 0.17, R 0.19, S 0.2, T 0.2

b 44 times

3 a 40 withdrawals 0.675 80 withdrawals 0.6375 120 withdrawals 0.633 160 withdrawals

0.63125 200 withdrawals 0.645

b 0.645, as this is from the most results

c 39 blues and 21 greens

4 a red 0.573, amber 0.183, green 0.244

b 82 more red lights than green

5 aNumberofquestions

Totalnumberwrong

Relativefrequencyto3significantfiguresifnecessary

10 4 0.4

20 8 0.4

30 11 0.367

40 15 0.375

50 18 0.36

60 20 0.333

70 22 0.314

b 0.314, because it is from the most results

c 47 wrong

6 a 1, 0.147; 3, 0.172; 5, 0.122; 7, 0.325; 9, 0.233

b i 23 or 24 timesii 37 times


1 0.31

2 124

3 18

4 0.27

5 2140

6 0.16

72325

8 320

9 0.36


1 a 1316

b12

2 0.45

3 a110

b1340

c1940

d15

e 2140

4 a 0.46b 0.68c 0.16d 0.19

5 a 59

b 79

c16

d536

6 a 0.42b 0.77c 0.11d 0.07e i 0.1

ii 0.3

7 a1730

b35

c320


d i17180

ii1790

8 a 0.37b 0.86c 0.126d 0.329e 0.441

9 a i1316

ii 916

b 34

10 a 0.54b 0.135c 0.055d 0.595e 0.865


1 a 625

b 1225

c 425

d 2125

2 a 521

b 1021

c 37

d 221

e 1921

3 a439

b1739

c613

4 a 136

b 16

c 14

d 89

5 a 0.6b 0.24c 0.48d 0.36

6 a 112

b 16

c 19

d 136

e49

f1136

7 a 649

b 149

c 349

d9

245

e87245

f1835

8 a3

200

b9

100

c325

d121400

e 125

f 399400

9 a 64295

b 11295

c 221295

d 249295


Exercise 47.5 (page 567) 1 a

pencil

69

pen

pen

pen

pencil

pencil

39

28

68

58

38

b 512

c 12

d 112

2 a

DVD

1015

CD

CD

CD

DVD

DVD

515

414

1014

914

514

b37

c1021

d221

3 a

black

2044

white

white

white

black

black

2444

2444

2044

2044

2444

b 36121

c 25121

d 60121

4 a red

yellow

white

724

1024

724

red

yellow

white

824

924

724

red

yellow

white

825

1025

725 red

yellow

white

824

1024

624

b 1475

c 320

d 3475

e 49100

5 a 0.25b red

green

amber

red

green

amber

0.55

0.55

0.55

0.2

0.25

0.2

0.25

0.2

0.25

red

green

amber

0.55

0.2

0.25 red

green

amber

c 0.3025d 0.5625e 0.32f 0.7975


6 a 14

b win

lose

draw

win

lose

draw

512

13

14

512

13

14

win

lose

draw

512

13

14 win

lose

draw

512

13

14

c 116

d 3572

e 59

7 a 5080

b1080

c2080

d adult

boy

girl

adult

boy

girl

5079

979

2079

5079

1079

1979

4979

1079

5079

adult

boy

girl

5080

1080

2080 adult

boy

girl

e 245632

f 87632

g 297316

8 a r

b

s

r

b

s

419

519

1019

419

619

919

319

619

1019

r

b

s

420

620

1020 r

b

s

b 619

c 395

d 91190

Exercise 47.6 (page 569) 1 a

not r

0.35r

pass

pass

not pass

not pass

0.65

0.84

0.16

0.05

0.95

b 0.3265


2 a

not t

310

t

d

d

not d

not d

710

13

23

45

15

b 5375

3 aw

not w

310

710

w

not w

310

710

w

not w

310

710

b 21100

c 3431000

4 a

not t

0.48t

m

m

not m

not m

0.52

0.4

0.6

0.35

0.65

b 0.374

5 a

not v

0.72v

c

c

not c

not c

0.28

0.12

0.88

0.72

0.28

b 0.288

6 a

not h

0.56h

w

w

not w

not w

0.44

0.72

0.28

0.18

0.82

b 0.4824


7 a

not s

0.48s

i

i

not i

not i

0.52

0.88

0.12

0.28

0.72

b 0.568

8 a

not w

38

w

w

w

not w

not w

58

56

16

25

35

b 179240

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